THE CLASSIFICATION OF HIGHLY CONNECTED MANIFOLDS IN

DIMENSIONS 7 AND 15.

Diarmuid J. Crowley

Submitted to the faculty of the University Graduate School

in partial fulfillment of the requirements for the degree Doctor of Philosophy in the Department of Mathematics Indiana University July 23rd 2001

Contents

Chapter 1. Introduction 1

Chapter 2. Preliminaries 13 1. Manifold essentials 13 2. Wall’s classification of the handlebodies L 16 3. Quadratic functions 19 4. Wall’s classification restated 23 5. Highly connected bordism 27 6. Decomposition of quadratic functions 30 7. Linking forms and Kronecker pairings 33 8. Wilkens’ classification of the manifolds P 40 9. Quadratic linking functions 43 10. Quadratic linking families 49 11. Categorical statements 63 12. The invariants s_{1}, s̄_{1} and μ 65

Chapter 3. Stable algebra of quadratic functions 70 1. Gluing and splitting nondegenerate quadratic functions 72 2. Stable equivalence of quadratic functions 81

Chapter 4. Classification of the manifolds P 87 1. Extensions of diffeomorphisms 87 2. Almost smooth classification of the manifolds P 89 3. The manifolds L(κ_{0} ∪_{θ} κ_{1}^{−}) and L_{0} ∪_{f^{o}} −L_{1} 91 4. Smooth classification of the manifolds P 98

Chapter 5. Linking forms and linking functions 102 1. Bilinear forms on finite abelian groups 102 2. Quadratic forms on finite abelian groups 107 3. Quadratic functions on finite abelian groups 111

Chapter 6. Classification up to homeomorphism and homotopy 123 1. Smoothing theory 123 2. The surgery exact sequence 127 3. Constructing smoothings in L and P 131 4. Smooth S^{3} bundles over S^{4} 143

Bibliography 147

CHAPTER 1 Introduction

An n-dimensional manifold M is called highly connected when all of its homotopy groups vanish below the middle dimension. That is, π_{i}(M) = 0 for all i < (n − 1)/2. We begin by briefly recalling the history of the classification of highly connected, closed, smooth manifolds in dimensions greater than 4. Throughout all manifolds will be smooth and oriented and all maps orientation preserving. In [Wa1] Wall built on the results of Smale, most notably the h-cobordism theorem, to classify highly connected even dimensional manifolds up to the addition of homotopy spheres. Later Wall [Wa3] presented a classification of highly connected odd dimensional manifolds up to the addition of homotopy spheres except in dimensions 5, 7 and 15. Barden, a student of Wall, solved the problem of the classification of simply connected 5manifolds [Ba]. In dimensions 7 and 15 Wilkens, another student of Wall, obtained results but they were not complete as we now explain [Wi1, Wi2].

Throughout this thesis P shall denote a highly connected, closed manifold of dimension (8j − 1) and L shall denote a handlebody by which we mean a highly connected, compact, even dimensional manifold with a nonempty and connected boundary. For every P there is a unique homotopy sphere such Σ_{P} (often the standard sphere) such that P #Σ_{P} is the boundary of a handlebody L and such that Σ_{P} bounds a Spin manifold with vanishing decomposable Pontryagin numbers and signature equal to zero (see Section 2.5). To each handlebody L, Wall associated the triple (H, λ, α), where H = H_{4j}(L; ℤ) is the middle homology group of L, λ : H × H → ℤ is the intersection form of L and α ∈ H^{4j}(L; ℤ) is the primary obstruction to the

1. INTRODUCTION 2

stable triviality of L. We identify H^{4j}(L; ℤ) with H^{∗}, the dual group of H, via the Kronecker pairing. Two triples (H_{0}, λ_{0}, α_{0}) and (H_{1}, λ_{1}, α_{1}) are isomorphic if there is an isomorphism Θ : H_{0} → H_{1} such that Θ^{∗}(α_{1}) = α_{0} and λ_{0} = λ_{1} ∘ (Θ × Θ). If the handlebodies L_{0} and L_{1} have associated triples (H_{0}, λ_{0}, α_{0}) and (H_{1}, λ_{1}, α_{1}) then Wall proved in [Wa1] that an isomorphism between these triples is realized by a diffeomorphism. Wilkens later proved that if f : ∂L_{0} → ∂L_{1} is a diffeomorphism between the boundaries of two handlebodies, then there are handlebodies V_{0} and V_{1} with standard spheres as boundary and a diffeomorphism g : L_{0}♮V_{0} → L_{1}♮V_{1} which extends f (here ♮ denotes the boundary connected sum). Now, the intersection form of a handlebody is nonsingular if and only if the handlebody cobounds a homotopy sphere. Thus the question of classifying the manifolds P up to connected with sum homotopy spheres becomes the question of the stable classification of triples (H, λ, α). Here we say that two triples (H_{0}, λ_{0}, α_{0}) and (H_{1}, λ_{1}, α_{1}) are stably equivalent if there are triples (H′_{0}, λ′_{0}, α′_{0}) and (H′_{1}, λ′_{1}, α′_{1}) such that λ′_{0} and λ′_{1} are nonsingular and (H_{0} ⊕ H′_{0}, λ_{0} ⊕ λ′_{0}, α_{0} ⊕ α′_{0}) is isomorphic to (H_{1} ⊕ H′_{1}, λ_{1} ⊕ λ′_{1}, α_{1} ⊕ α′_{1}).

To classify the manifolds P up to almost diffeomorphism Wilkens associated to each P the triple (G, b, β) where G = H^{4j}(P; ℤ), b is the linking form of P which is a nonsingular symmetric bilinear function b : T G × T G → ℚ/ℤ where T G is the torsion subgroup of G and β ∈ G is the primary obstruction to the triviality of the stable tangent bundle of P. We shall call this triple the Wilkens triple or Wilkens invariants of P. Two triples (G_{0}, b_{0}, β_{0}) and (G_{1}, b_{1}, β_{1}) are isomorphic if there is an isomorphism θ : G_{0} → G_{1} such that θ(β_{0}) = β_{1} and b_{0} = b_{1} ∘ (θ × θ). The only constraint on the triples which are realized by highly connected 7 and 15 manifolds is that β = 2γ for some γ ∈ G. In higher dimensions all triples are realized. If P = ∂L and (H, λ, α) is the triple associated to L then (H, λ, α) induces (G, b, β) in a natural way. Wilkens was able to prove that if the group G contains no 2-torsion

1. INTRODUCTION 3

then any two highly connected (8j − 1)-manifolds, j = 1, 2, with isomorphic triples are diffeomorphic after the addition of homotopy spheres. When G contains 2 torsion and b is not decomposable as the sum of smaller nontrivial linking forms, Wilkens showed that up to connected sum with homotopy spheres, there were at most two highly connected (8j − 1)-manifolds, j = 1, 2 with isomorphic triples. For a precise statement of Wilkens’ results we refer the reader to Section 2.8. A ℤ/2-ambiguity associated to groups of even order points to the need for a quadratic refinement of the linking form b. Indeed Wall had left the classification unsolved in dimensions 7 and 15 because the existence of bundles of Hopf-invariant one over the 4-sphere and 8-sphere meant that his definition of the quadratic refinement of the linking form of a highly connected manifold did not work. Wilkens, however, did not interpret his results on the stable algebra of triples (H, λ, α) in terms of quadratic refinements of the induced linking forms.

Now let G be a finite abelian group. A quadratic linking function on G is a function q : G → ℚ/ℤ such that

q(x + y) = q(x) + q(y) + b(x, y)

for every x, y ∈ G and for some linking form, b, on G. If q is homogeneous, that is q(−x) = q(x) for every x ∈ G, then q is called a quadratic linking form. We let Q(b) denote the set of quadratic linking functions refining a linking form b. There is a free action of G on Q(b) defined, for a ∈ G and q ∈ Q(b), by a · q = q_{a} where for every x ∈ G,

q_{a}(x) = q(x) + b(x, a).

A nondegenerate, even bilinear form λ : H × H → ℤ on the free abelian group H induces a quadratic linking form, q(λ), on G = Cok(λ̂) where λ̂ : H → H^{∗} is the adjoint of λ (see Definition 2.32). Both Durfee [Du1, Du2] and Wall [Wa4] proved that if λ_{0} and λ_{1} are nondegenerate even bilinear forms then λ_{0} and λ_{1} are stably

1. INTRODUCTION 4

equivalent if and only if q(λ_{0}) ≅ q(λ_{1}), where stable equivalence means isomorphic after the addition of nonsingular even bilinear forms. However, they did not consider the question of the stable equivalence of the triples (H, λ, α). In particular in dimensions 8 and 16 the triples (H, λ, α) arising from handlebodies need not contain even intersection forms. So these algebraic results were confined to topological applications in dimensions other than 7 and 15.

Thus the question of the classification of 2-connected 7-manifolds and 6-connected 15-manifolds stood where Wilkens left it for thirty years.^{1} Then Grove and Ziller showed that the total spaces of S^{3}-bundles over S^{4} admit metrics of non-negative sectional curvature and they called for a classification of these total spaces [GZ, Problem 5.2]. Of course such total spaces are 2-connected 7-manifolds. This problem was solved in [CE] where we showed that the invariants s_{1} and s̄_{1} defined in [KS] ^{2} resolved the ambiguity in Wilkens’ classification. Specifically we showed that if P_{0} and P_{1} are the total spaces of S^{3}-bundles over S^{4} which do not admit a cross section then P_{0} is homeomorphic (resp. diffeomorphic) to P_{1} if and only if their Wilkens invariants are isomorphic and s̄_{1}(P_{0}) = s̄_{1}(P_{1}) (resp. s_{1}(P_{0}) = s_{1}(P_{1})). We shall prove that the same statement holds for all 2-connected 7-dimensional rational homology spheres.

Now let L be a handlebody of dimension 8 or 16 with associated triple (H, λ, α). From the point of view of Durfee’s and Wall’s results above the difficulty in dimensions 8 and 16 is that λ need not be an even form which is an immediate consequence of the existence of bundles of Hopf invariant one. However, this is compensated by the fact that the existence of bundles of Hopf invariant one ensures that the tangential invariant, α, of L reduces mod 2 to either w_{4} or w_{8}, the fourth or eight Stiefel-Whitney

1This is not quite correct. Madsen, Taylor and Williams [MTW] claimed that Wilkens’ ambiguous pairs are homeomorphic but this is not so. However more on this in Section 6.4.

2We explain the invariants s_{1} and s̄_{1} in detail in Section 2.12. They are defined in all dimensions 4k − 1 and in dimensions 7 and 15, s_{1} coincides with the Eels-Kuiper μ-invariant [EK].

1. INTRODUCTION 5

class. Thus α is characteristic for λ by which we mean that the quadratic function κ(λ, α) defined by

κ(λ, α)(v) = λ(v, v) + α(v)

takes on even values for every v ∈ H. When λ is nondegenerate this fact allows the definition of a quadratic linking function on the cokernel of λ̂ as follows. For every x ∈ H^{∗} there is a nonzero integer r and an element v ∈ H such that λ̂(v) = rx. If [x] denotes the image of x in Cok(λ̂) then

q^{c}(κ(λ, α))([x]) := \frac{1}{2r}(x(v) + α(v)) mod ℤ.

We note that q^{c}(κ(λ, α)) is not necessarily homogeneous. Let P be a highly connected (8j − 1)-manifold, j = 1, 2. Recall that there is a homotopy sphere Σ_{P} such that P #Σ_{P} = ∂L where L is a handlebody. Suppose that L defines the triple (H, λ, α). Then, identifying H^{4j}(P) and H^{4j}(P #Σ_{P}) if necessary, Cok(λ̂) = H^{4j}(P). If P is a rational homology sphere then H^{4j}(P) is finite and λ is nondegenerate so we may define the quadratic linking function of P by

q^{c}(P) := q^{c}(κ(λ, α)).

We shall show that q^{c}(P) is independent of the choice of handlebody and refines b, the linking form of P. Also we show that for any quadratic linking form q which refines b there is an element a ∈ G such that q^{c}(P) = q_{a} and that the Wilkens invariant of P is (G, b, 2a).

As we first classify the manifolds P up connected sum with homotopy spheres we introduce the notion of an almost diffeomorphism. A homeomorphism f : M_{0} → M_{1} between smooth manifolds is an almost diffeomorphism if it is a diffeomorphism except at a finite number of singular points. Two manifolds are called almost diffeomorphic when there is an almost diffeomorphism between them. Note that M_{0} and M_{1} are

1. INTRODUCTION 6

almost diffeomorphic if and only if there is a single homotopy sphere Σ and a diffeomorphism f′ : M_{0}#Σ → M_{1}. Given any degree one map f : P_{0} → P_{1} between two highly connected (8j − 1)-manifolds we define the induced Umker map on cohomology, f_{!} : H^{4j}(P_{0}) → H^{4j}(P_{1}), where for x ∈ H^{4j}(P_{0}), f_{!}(x) := f_{∗}(x ∩ [P_{0}])/(P_{1}). Here [P_{0}] ∈ H_{8j}(P_{0}) and (P_{1}) ∈ H^{8j}(P_{1}) are the orientation classes of P_{0} and P_{1} and ∩ and / denote the cap and slant products respectively.

Theorem A. Let j = 1 or 2. For every finite group G, every linking form b : G × G → ℚ/ℤ and every quadratic linking function q ∈ Q(b), there is a highly connected (8j − 1)-dimensional rational homology sphere, P, such that q^{c}(P) ≅ q. If P_{0} and P_{1} are two highly connected (8j − 1)-dimensional rational homology spheres with quadratic linking functions q^{c}(P_{0}) and q^{c}(P_{1}) and Wilkens invariants (G_{0}, b_{0}, β_{0}) and (G_{1}, b_{1}, β_{1}), then,

1. there is an almost diffeomorphism f : P_{0} → P_{1} such that f_{!} = θ : H^{4j}(P_{0}) ≅ H^{4j}(P_{1}) if and only if q^{c}(P_{0}) = q^{c}(P_{1}) ∘ θ,

2. there is a diffeomorphism f : P_{0} → P_{1} such that f_{!} = θ : H^{4j}(P_{0}) ≅ H^{4j}(P_{1}) if and only if q^{c}(P_{0}) = q^{c}(P_{1}) ∘ θ, s_{1}(P_{0}) = s_{1}(P_{1}) and Σ_{P_{0}} ≅ Σ_{P_{1}},

3. P_{0} and P_{1} are almost diffeomorphic if and only if (G_{0}, b_{0}, β_{0}) ≅ (G_{1}, b_{1}, β_{1}) and s̄_{1}(P_{0}) = s̄_{1}(P_{1}),

4. P_{0} and P_{1} are diffeomorphic if and only if (G_{0}, b_{0}, β_{0}) ≅ (G_{1}, b_{1}, β_{1}), s_{1}(P_{0}) = s_{1}(P_{1}) and Σ_{P_{0}} ≅ Σ_{P_{1}},

5. in dimension 7, “almost diffeomorphism” may be replaced by “homeomorphism” in 1,

6. in dimension 7 there is a homotopy equivalence f : P_{0} → P_{1} such that f_{!} = θ : H^{4j}(P_{0}) ≅ H^{4j}(P_{1}) if and only if q^{c}(P_{0})_{12a} = q^{c}(P_{1}) ∘ θ for some a ∈ G_{0}.

1. INTRODUCTION 7

Remark 1.1. The proof of Theorem A is distributed throughout the thesis as follows. Part 1 is a special case of Theorem B which is proven in Theorem 4.2. Part 2 is a special case of Theorem 4.9. Parts 3 and 4 are equivalent to Corollary 5.24. Part 5 is proven in Theorem 6.1 and Part 6 is a special case of Theorem 6.11.

Remark 1.2. Note that we do not follow the classical surgery path of first classifying the manifolds P up to homotopy and then up to homeomorphism. Nor do we use Kreck’s [Kr2] modified surgery. Rather, we use Wall’s [Wa1] classification on the handlebodies L and consider more closely the algebraic structures particular to dimensions 7/8 and 15/16 which are induced on the boundaries P.^{3} In Chapter 6 we use surgery theory and the topological classification of part 5 to deduce the homotopy classification.

Remark 1.3. Part 4 of this theorem extends Theorem 6 of [Kr2] to cases where H^{4}(P_{i}) is not generated by β_{i}. After completing this thesis we discovered that F. Bermbach had already undertaken such an extension in [Be]. However, Bermbach’s general theorem on the classification of simply-connected 7-manifolds uses a Wall style homogeneous quadratic refinement which is not easy to calculate. Moreover, he does not apply this theorem to the classification of 2-connected 7-manifolds but in applications he is exclusively concerned with certain homogeneous spaces having π_{2} ≅ ℤ^{2}.

We now proceed to the general case where G = H^{4j}(P) might have a nontrivial free subgroup. If G is itself free then the classification problem is easily solved by Wall’s classification of handlebodies.

Proposition 1.4. Let P_{0} and P_{1} be highly connected (8j − 1)-manifolds with Wilkens invariants (G_{0}, b_{0}, β_{0}) and (G_{1}, b_{1}, β_{1}) and suppose that G_{0} and G_{1} are free

3This is the route suggested by Wall just after Theorem 8 in [Wa3] (and followed by Wilkens).

1. INTRODUCTION 8

groups. Then there is an almost diffeomorphism f : P_{0} → P_{1} such that f_{!} = θ : G_{0} ≅ G_{1} if and only if θ(β_{0}) = β_{1}. Moreover, if G_{0} ≅ ℤ^{l} then P_{0} is diffeomorphic to P′_{0}#(#_{i=1}^{l−1}(S^{4j−1} × S^{4j})) where H^{4j}(P′_{0}) ≅ ℤ.

If, however, G = H^{4j}(P) has a nontrivial torsion subgroup T G and a nontrival free subgroup then the classification becomes a more complicated matter because in this case a highly connected (8j − 1)-manifold P, (j = 1, 2), does not necessarily define a unique quadratic linking function on T G. We are therefore lead to the notion of a quadratic linking family which we now explain. Firstly, let τ_{G} denote the canonical homomorphism

\begin{aligned}τ_{G} : & G & → & G^{∗∗} \\ & g & ↦ & (x ↦ x(g)) \end{aligned}

and let Sec(τ_{G}) denote the set of homomorphisms Φ : G^{∗∗} ↪ G such that τ_{G} ∘ Φ = Id_{G^{∗∗}}. A quadratic linking family is a triple (G, q^{∗}, β) consisting of the finitely generated abelian group G, an element β ∈ G and a function q^{∗} : Sec(τ_{G}) → Q(b) for some linking form on T G. The superscript ∗ takes the values c for characteristic and ev for even. When j is greater than 2, highly connected (8j − 1)-manifolds define even quadratic linking families on their 4jth cohomology groups. The function q^{ev} is always a constant function whose value is a quadratic linking form. Thus an even quadratic linking family is equivalent to a triple (G, q, β) where q is a quadratic linking form on T G. However, in dimensions 7 and 15, highly connected manifolds define characteristic quadratic linking families whose function q^{c} may be far from constant. We refer the reader to Section 2.10 for the precise definitions of quadratic linking families and isometries between them. A triple (H, λ, α) (where λ may now be a degenerate bilinear form) induces a quadratic linking family, Q(λ, α) on Cok(λ̂) = G. Given P, we define Q(P) to be Q(λ, α) where P #Σ_{P} = ∂L for a handlebody with triple (H, λ, α) (again see Section 2.10).

1. INTRODUCTION 9

Theorem B. Let j = 1 or 2. For every finitely generated abelian group G and every characteristic quadratic linking family Q defined on G, there is a highly connected (8j − 1)-manifold P such that Q(P) ≅ Q. The quadratic linking family Q(P) of a highly connected (8j −1)-manifold P is a complete invariant of almost diffeomorphisms. That is, if P_{0} and P_{1} are two highly connected (8j − 1)-manifolds then there is an almost diffeomorphism f : P_{0} → P_{1} such that f_{!} = θ : H^{4j}(P_{0}) ≅ H^{4j}(P_{1}) if and only if θ defines an isometry of quadratic linking families from Q(P_{0}) to Q(P_{1}).

For comparison we present the analogous theorem in higher dimensions which follows rapidly from a compilation of Wall’s results on highly connected manifolds.

Theorem 1.5 (Wall [Wa1, Wa2, Wa3, Wa4]). Let j > 2. For every finitely generated abelian group G and every even quadratic linking family Q on G there is a highly connected (8j − 1)-manifold P such that Q(P) ≅ Q. The quadratic linking family Q(P) of a highly connected (8j − 1)-manifold P is a complete invariant of almost diffeomorphisms.

The remainder of this thesis is organized as follows. In Chapter 2 we develop the necessary preliminaries. We first summarize Wall’s classification of the handlebodies L and rephrase it in terms of quadratic functions. Here a quadratic function on a free abelian group H is a function κ(λ, α) : H → ℤ such that κ(v) = λ(v, v) + α(v) for λ a symmetric bilinear form on H and α a linear function on H. We then show that after the addition of a homotopy sphere, every manifold P bounds a handlebody L and present Wilkens’ Theorem (2.24) on the extension of diffeomorphisms of the manifolds P to cobounding handlebodies. At this stage it is clear that the almost diffeomorphism classification of the manifolds P is equivalent to the stable classification of quadratic functions where “stable” means “modulo nonsingular quadratic functions”. We therefore devote the sections which follow to the algebraic invariants

1. INTRODUCTION 10

that a quadratic function, κ(λ, α), induces on its cokernel, G = Cok(λ̂), which are stably invariant. These invariants are, in increasing order of sophistication: a linking form on the torsion subgroup of G, a quadratic refinement of this linking form and, in the case where κ in degenerate, a quadratic linking family. In the final two sections of the chapter we provide categorical statements of our classification and define the invariants s_{1} and s̄_{1}.

Chapter 3 is an entirely algebraic chapter concerning the stable classification of quadratic functions. We prove that quadratic functions are stably classified by the quadratic linking families which they induce. We first consider the case of nondegenerate quadratic functions where the key idea is that quadratic linking functions behave like algebraic boundaries to the quadratic functions which induce them. In particular, the Gluing Lemma 3.3.6 states that if quadratic functions κ_{0} and κ_{1} induce quadratic linking functions q_{0} and q_{1}, then an isometry, θ : q_{0} ≅ q_{1}, may be used to glue κ_{0} to κ_{1} and form a nonsingular quadratic function which we denote κ_{0} ∪_{θ} κ_{1}^{−}. This procedure is a precise algebraic analogue of gluing two handlebodies together along a diffeomorphism of their boundaries. We are then able to mimic the proof of Wilkens’ Theorem 2.24 in the algebraic setting and show that the isometry θ : q_{0} → q_{1} above is induced by an isometry of quadratic functions Θ : κ_{0} ⊕ κ_{0} ∪_{θ} κ_{1}^{−} → κ_{1} ⊕ μ where μ is a nonsingular quadratic function. Taking a little more care we seen that the somewhat complex notion of a quadratic linking family is the required algebraic object to generalize this result for degenerate quadratic functions. We note that our results in this chapter generalize the results of Wall and Durfee on the stable classification of nondegenerate even quadratic forms (in a natural way) to the stable classification of nondegenerate characteristic quadratic functions.

In Chapter 4 we prove our central classification results, Theorem B and Theorem 1.5, by applying Wilkens’ Theorem 2.24 to the algebraic results obtained in Chapter

1. INTRODUCTION 11

3. As well as completing the classification left open by Wilkens’ this gives a new and simpler proof of Wall’s Theorem 1.5. In Section 3 we use the h-cobordism theorem to show that the algebraic Gluing and Splitting Lemmas (3.3.6 and 3.3.10) from Chapter 3 are mimicked precisely in the world of handlebodies. In the case of rational homology spheres, this gives a second proof of Theorem B and Theorem 1.5 which is independent of Theorem 2.24. In the last section of the chapter we present the smooth classification of highly connected (8j − 1)-manifolds.

In Chapter 5 we review the classification of linking forms and quadratic linking forms on finite abelian groups due to Kawauchi and Kojima [KK] and Nikulin [Ni] respectively and we then extend Nikulin’s classification of quadratic linking forms to the classification of quadratic linking functions. Specifically, let G be a finite abelian group and let q be a quadratic linking function on G which refines the linking form b(q). There is a quadratic linking form q^{o} such that q = q_{a}^{o} for a ∈ G. If we define β(q) := 2a then the pair (b(q), β(q)) is an invariant of q (see 2.38). In addition to the invariant (b(q), β(q)) there is a Kervaire-Arf invariant, K(q) ∈ ℚ/ℤ, defined for each quadratic linking function q via a Gauss sum (see Section 5.3). Our classification result is the following

Theorem 5.22. Let q_{0} and q_{1} be quadratic linking functions over a finite abelian group G. Then q_{0} is isometric to q_{1} if and only if (b(q_{0}), β(q_{0})) ≅ (b(q_{1}), β(q_{1})) and K(q_{0}) = K(q_{1}).

If q^{c}(P) is the quadratic linking function of a highly connected (8j − 1)-manifold (j = 1, 2), we also show that K(q(P)) = −s̄_{1}(P). Hence Theorem 5.22 is the algebraic counter-part to 3 of Theorem A.

In Chapter 6 we use smoothing theory and surgery theory to deduce respectively the topological and homotopical classification of highly connected 7-manifolds from

1. INTRODUCTION 12

the almost diffeomorphism classification of these manifolds. We conclude by applying our results to give the classification of the total spaces of 3-sphere bundles over the 4-sphere which appears in [CE].

CHAPTER 2 Preliminaries

1. Manifold essentials

Throughout this thesis all manifolds shall be compact and oriented and all maps between manifolds shall be assumed orientation-preserving. If M is a manifold then −M shall denote the manifold with opposite orientation. Unless otherwise noted manifolds shall also be smooth. The letter L shall always denote a 2k-dimensional handlebody by which we mean a manifold obtained by attaching some number of k-handles to the 2k-ball. We review the definition and theory of handlebodies in Section 2. For all k > 2 every highly connected 2k-manifold with a single boundary component is diffeomorphic to a handlebody. The symbol ∂ shall be used to indicate the operation of taking the boundary of a manifold. The boundary of every handlebody is a highly connected manifold of dimension (2k − 1). The symbol M_{0}#M_{1} denotes the (interior) connect sum of the manifolds M_{0} and M_{1}. The symbol L_{0}♮L_{1} denotes the boundary connect sum of L_{0} and L_{1}. We refer the reader to [KM] for definitions. It is a consequence of these definitions that

∂(L_{0}♮L_{1}) = ∂L_{0}#∂L_{1}.

A simply connected h-cobordism is a simply connected compact (n + 1)-manifold W whose boundary has two components, ∂W = M_{0} ⊔ −M_{1}, such that each inclusion M_{i} ↪ W is a homotopy equivalence for i = 0, 1. The h-cobordism theorem of Smale asserts that when n > 4 every simply connected h-cobordism is diffeomorphic to the product M_{0} × [0, 1] via a diffeomorphism that is the identity restricted to M_{0}.

1. MANIFOLD ESSENTIALS 14

A homotopy sphere is a manifold which is homotopy equivalent to the standard sphere. We shall always denote homotopy spheres by the letter Σ. In dimensions higher than 4 Kervaire and Milnor [KM] famously classified the groups of diffeomorphism classes of homotopy spheres under the operation of connected sum. The group of n-dimensional homotopy spheres is denoted by Θ_{n} and the subgroup of those spheres bounding parallelizable manifolds by bP_{n+1}. There is an exact sequence

0 ⟶ bP_{n+1} ⟶^{i} Θ_{n} ⟶ Cok(J_{n})

where Cok(J_{n}) denotes the cokernel of the J-homomorphism, J_{n} : π_{n}(O) → π_{n}^{S} (see [Wh] pg 502-4). When n = 4k − 1, Θ_{4k−1} ≅ bP_{4j} ⊕ Cok(J_{n}) and Brumfiel defined a splitting s : Θ_{4k−1} → bP_{4k} of i as follows. The group bP_{4k} ≅ ℤ/|bP_{4k}| is cyclic (of order determined by the kth Bernoulli number). Brumfiel [Bru] showed that every homotopy sphere Σ ∈ Θ_{4k−1} bounds a spin manifold W with vanishing decomposable Pontryagin numbers. The homomorphism s is defined by Σ ↦ σ(W)(mod|bP_{4k}|) where σ(W) denotes the signature of the 4k-manifold W. For the dimensions of primary concern in this thesis, namely 7 and 15, the groups of exotic spheres are:

Θ_{7} = bP_{8} ≅ ℤ/28 Θ_{15} ≅ bP_{16} ⊕ ℤ/2 ≅ ℤ/8128 ⊕ ℤ/2.

Let M_{0} and M_{1} be smooth manifolds. A homeomorphism f : M_{0} → M_{1} which is a diffeomorphism except at a finite number of singular points shall be called an almost diffeomorphism.

Proposition 2.1. Let f : M_{0} → M_{1} be an almost diffeomorphism of n-dimensional manifolds. Then there is an embedded disc D^{n} ↪ M_{0} containing all the singular points of f, a homotopy sphere Σ and a diffeomorphism f′ : M_{0}#Σ → M_{1} such that f′|_{M_{0}−D^{n}} = f|_{M_{0}−D^{n}}. (Here it is understood that the connected sum of M_{0} and Σ takes place inside D^{n}.)

1. MANIFOLD ESSENTIALS 15

Proof. Let p_{1}, … p_{l} be the singular points of f and let I_{i} : D_{i}^{n} ↪ M_{0} be l pairwise disjoint embeddings of discs with I_{i}(0) = p_{i}. The image of each D_{i}^{n} under f is a subset of M_{1} homeomorphic to the n-disc and restricting the smooth structure of M_{1} to f(D_{i}^{n}) yields a smooth manifold which is diffeomorphic to the n-disc. So let J_{i} : D_{i1}^{n} be embeddings of discs in M_{1} such that J_{i}(D_{i1}^{n}) = f(I_{i}(D_{i}^{n})) and J_{i}(0) = f(p_{i}). There are then diffeomorphisms f_{i} : (D^{n}−0) → (D_{i1}^{n}−0) defined by f_{i} = J_{i}^{−1}∘f ∘I_{i}|_{D_{i}^{n}−0} and we construct the homotopy spheres Σ_{i} := −D_{i}^{n} ∪_{f_{i}} D_{i1}^{n}. Using the obvious embedding of −D_{i}^{n} into Σ_{i} and I_{i} we define the connected sum M_{0}#Σ_{1}# … #Σ_{l}. There is then a diffeomorphism

f′′ : M_{0}#Σ_{1}# … #Σ_{l} → M_{1}
M_{0} − {p_{0}, …, p_{l}} ∋ x ↦ f(x)
D_{i1}^{n} ∋ x  ↦ J_{i}(x).

Now let Σ = Σ_{1}# … #Σ_{l} and let D^{n} be a disc in M_{0} containing ∪_{i=1}^{l}D_{i}^{n} in its interior. There is a diffeomorphism f′′′ : M_{0}#Σ → M_{0}#Σ_{1}# … #Σ_{l} such that f′′′|_{M^{0}−D^{n}} = Id_{M^{0}−D^{n}}. We may then take f′ to be the diffeomorphism f′′ ∘ f′′′.

Now let f_{0} : M_{0} → M_{1} and f_{1} : M_{1} → M_{2} be a pair of almost diffeomorphisms with singular points p_{0}, …, p_{l_{0}} and q_{0}, …, q_{l_{1}} respectively. Their composition, f_{1} ∘ f_{0} is a homeomorphism with singular points p_{0}, …, p_{l_{0}}, f_{0}^{−1}(q_{0}), …, f_{0}^{−1}(q_{l_{1}}) (some of which may be repeated) and hence an almost diffeomorphism. Thus, for any class of smooth manifolds there is a corresponding category whose morphisms are almost diffeomorphisms. We work primarily in the almost diffeomorphism category because this is the level at which the algebra of the manifolds L and P is the simplest.

By an almost smooth manifold we shall mean a P L n-manifold, M, with a smooth structure except at a finite number of singular points, p_{1}, …, p_{l}. We may assume that there are mutually disjoint discs D_{i}^{n} with p_{i} ∈ int(D_{i}^{n}). The manifold M − (⊔_{i=1}^{l}D_{i}^{n}) is a smooth manifold with boundary l disjoint homotopy spheres Σ_{1}, …, Σ_{l} and the

2. WALL’S CLASSIFICATION OF THE HANDLEBODIES L 16

P L structure on D_{i}^{n} is obtained by coning on Σ_{i}. Almost smooth manifolds shall arise in this thesis in two ways. Firstly, we shall sometimes consider a handlebody L with boundary an exotic sphere Σ and form the almost smooth manifold L ∪ D^{n} with singular point at the center of the disc D^{n}. At other times we shall consider two handlebodies L_{0} and L_{1} and an almost diffeomorphism f : ∂L_{0} → ∂L_{1}. The adjunction space L_{0} ∪_{f} −L_{1} has the structure of an almost smooth manifold with singular points corresponding to the singular points of f. An almost smooth h-cobordism between closed simply connected smooth manifolds M_{0} and M_{1} is a compact almost smooth manifold W with boundary M_{0}⊔−M_{1}, singular points contained in W −(M_{0}⊔M_{1}) and with the inclusions M_{i} ↪ W homotopy equivalences. We now demonstrate that the boundaries of such an almost smooth h-cobordism are almost diffeomorphic. Let the singular points of W be p_{0}, …, p_{n} and suppose that they are contained in pair-wise disjoint closed discs D_{i}^{n} whose boundaries are diffeomorphic to Σ_{i} = ∂D_{i}^{n} with the smooth structure they inherit from W. In addition let γ_{i} : [0, 1] × int(D^{n−1}) ↪ W be pair-wise disjoint smooth embeddings such that γ_{i}^{−1}(D_{i}^{n}) = 1 × int(D_{i}^{n−1}) and γ_{i}^{−1}(∂W) = 0 × int(D_{i}^{n−1}) ⊂ M_{0}. The manifold W − (⨆_{i} Im(γ_{i})) is a smooth simply connected h-cobordism between with boundary M_{0}#(Σ_{1}# … #Σ_{n})⊔−M_{1} and hence M_{0} and M_{1} are almost diffeomorphic by the h-cobordism theorem.

2. Wall’s classification of the handlebodies L

We adopt the notation of [Sm] and consider manifolds L = L(D^{2k}, ϕ_{1}, …, ϕ_{l}; k) where ϕ_{i} : (∂D^{k}×D^{k})_{i} → D^{2k} are embeddings and L is obtained from D^{2k}⊔⊔_{i=1}^{l}(D^{k}× D^{k})_{i} by identifying points using the ϕ_{i} and then smoothing corners. The embedding

ϕ = ⊔_{i=1}^{l}ϕ_{i} : ⊔_{i=1}^{l}(S^{k−1} × D^{k})_{i} ↪ ∂D^{2k}

is called a presentation of L. Smale denoted the set of manifolds formed in this way by H(2k, l, k) and defined H(k) := ⋃_{l=0}^{∞} H(2k, l, k). The isotopy extension theorem

2. WALL’S CLASSIFICATION OF THE HANDLEBODIES L 17

entails that the diffeomorphism type of L(D^{2k}, ϕ_{1}, …, ϕ_{l}; k) is determined by the isotopy class of the embedding ϕ. Two pieces of data determine the isotopy class of ϕ. Firstly if one considers ϕ restricted to the attaching spheres of each handle one obtains a link

ϕ̄ = ⊔_{i=1}^{l}ϕ̄_{i} : ⊔_{i=1}^{l}(S^{k−1} × 0)_{i} ↪ S^{2k−1}.

When k > 2, Smale proved in [Sm] that the isotopy class of ϕ̄ is completely determined by the linking numbers of the attaching spheres. Setting S_{i}^{k−1} = (S^{k−1} × 0)_{i}, the linking number of ϕ̄_{i}(S_{i}^{k−1}) with ϕ̄_{j}(S_{j}^{k−1}) is c_{ij} where ϕ̄_{i}(S_{i}^{k−1}) is homologous to c_{ij} times ϕ(0 × ∂D^{k})_{j} in H_{k−1}(∂D^{2k−1} − ϕ̄_{j}(S_{j}^{k−1})). Secondly, for each ϕ_{i} there are framing invariants, ᾱ_{i} ∈ π_{k−1}(SO(k)), which are defined as follows. The subspace N := ϕ_{i}(S^{k−1} × D^{k}) is a tubular neighborhood of ϕ̄_{i}(S_{i}^{k−1}) in S^{2k−1}. There is a framing of N, ψ_{i} : N ≅ ϕ̄_{i}(S_{i}^{k−1}) × D^{k}, uniquely defined up to homotopy, such that ψ extends to a framing of the normal bundle of a k-disc which is embedded in D^{2k} and which bounds ϕ̄_{i}(S_{i}^{k−1}). After isotopy of ϕ_{i} we may assume that the composition ψ_{i} ∘ ϕ_{i} has the form

\begin{aligned}ψ_{i} ∘ ϕ_{i} : & (S^{k−1} × D^{k})_{i} & → & ϕ̄_{i}(S_{i}^{k−1}) × D^{k} \\ & (x, y) & ↦ & (ϕ̄_{i}(x), a(ϕ_{i})(x)(y)) \end{aligned}

for some map a(ϕ_{i}) : S^{k−1} → SO(k). While the map a(ϕ_{i}) depends upon the choices made the homotopy class of a(ϕ_{i}), which we denote by ᾱ_{i}, does not.

Lemma 2.2 (Wall [Wa1]). The invariants c_{ij}, (i ≠ j) and ᾱ_{i} form a complete set of isotopy invariants of the presentation ϕ.

Wall then proceeds to identify abstract invariants for handlebodies L which are counter parts to the invariants of a presentation but which do not depend upon giving a presentation for L. Viewed abstractly, the manifold L has invariants a free abelian group H = H_{k}(L), an intersection pairing λ : H × H → ℤ, and a map

2. WALL’S CLASSIFICATION OF THE HANDLEBODIES L 18

ᾱ : H → π_{k−1}(SO(k)). Geometrically λ is defined as follows. By the Hurewicz theorem H_{k}(L) ≅ π_{k}(L) and so every homology class v ∈ H is represent by a map v̄ : S^{k} → L. By the embedding theorems of Whitney this map may be chosen to be an embedding. Given v, w ∈ H we may assume further that the embeddings v̄ : S^{k} → L and w̄ : S^{k} → L are in general position and so intersect at isolated points. The integer λ(v, w) is the signed sum of these intersection points where the sign comes from comparing the orientation of L with orientation induced by v̄ and w̄ at a particular intersection point. The map ᾱ evaluated on a homology class v is given by mapping the normal bundle of v̄ : S^{k} ↪ L to the homotopy class of its clutching function which is an element of π_{k−1}(SO(k)). If ∂ι_{k} denotes the element of π_{k−1}(SO(k)) corresponding to the tangent bundle of the k-sphere, then Wall proves:

Lemma 2.3 (Wall [Wa1]).

ᾱ(v + w) = ᾱ(v) + ᾱ(w) + λ(v, w) · ∂ι_{k}.

This allows us to translate between the abstract invariants of L and the invariants of the presentation for L = L(D^{2k}, f_{1}, …, f_{n}; k). If one attaches a disc D_{i}^{k} inside D^{2k} to the core of each handle (D^{k} × 0)_{i} then one obtains a k-sphere v̄_{i} which one may assume is embedded in L. The homology classes v_{i} corresponding to these embedded spheres form a basis for H = H_{k}(L). Moreover the spheres v̄_{i} intersect only inside D^{2k} and their intersection number λ(v_{i}, v_{j}) is coincides with linking number c_{ij}. The self linking numbers λ(v_{i}, v_{i}) are given by the Euler numbers E(ᾱ_{i}) of the framings from the presentation invariants. Finally the framings ᾱ_{i} are evidently given by ᾱ(v_{i}) and Lemma 2.3 ensures that the framings ᾱ_{i} determine ᾱ. Wall is able to complete this translation between presentations and the abstract point of view by proving

Theorem 2.4 (Theorem 1 [Wa1]). Suppose L ∈ H(k), and choose a basis for H_{k}(L). Then L has a presentation corresponding to this basis.

3. QUADRATIC FUNCTIONS 19

For much of the rest of this thesis k shall be equal to 4j and we shall be concerned almost exclusively with 8j-dimensional handlebodies and more particularly with the boundaries of 8j-dimensional handlebodies. It will be algebraically convenient to consider the stabilization of the tangential invariant ᾱ which is α := S ∘ ᾱ, where S : π_{4j−1}(SO(4j)) → π_{4j−1}(SO) ≅ ℤ is the stabilization map. Since S(∂ι_{k}) = 0, Lemma 2.3 implies that α : H → ℤ is a homomorphism. In fact, Kervaire computed that α = a_{j}(2j − 1)!p_{j} where p_{j} denotes the jth Pontryagin class and a_{j} = 1 if j is even and 2 if j is odd, [Ke]. Note that we lose nothing by considering α over ᾱ since the unstable part of ᾱ(v) is given by the Euler number of the normal bundle of v̄ and this is equal to λ(v, v). Hence ᾱ may be recovered from α and λ. We conclude this section with a summary in the form of a Corollary.

Corollary 2.5. Let k > 2 and let L be a highly connected 2k-manifold with a non-empty and connected boundary. Then L is diffeomorphic to a handlebody. If L_{0} and L_{1} are two handlebodies of dimension 2k with invariants (H_{0}, λ_{0}, α_{0}) and (H_{1}, λ_{1}, α_{1}) then there is diffeomorphism f : L_{0} → L_{1} such that f_{∗} = Θ if and only if there is an isomorphism Θ : H_{0} → H_{1} such that λ_{1}(Θ(v), Θ(w)) = λ_{0}(v, w) and α_{1}(Θ(v)) = α_{0}(v) for all v, w ∈ H_{0}.

Remark 2.6. Note that the P L-invariance of rational Pontryagin classes (see [MS] §20) entails that the words words “diffeomorphic” and “diffeomorphism” in Corollary 2.5 may be replaced by “P L-homeomorphic” and“P L-homeomorphism”. Novikov’s result on the topological invariance of the rational Pontryagin classes entails that they may be replaced by “homeomorphic” and “homeomorphism”.

3. Quadratic functions

In the previous section we saw Wall’s classification of the handlebodies L via their intersection form λ(L) : H_{4j}(L) × H_{4j}(L) → ℤ and their stable tangential

3. QUADRATIC FUNCTIONS 20

invariant α(L) ∈ H^{4j}(L). In this section we give an abstract summary of the algebraic invariants of L.

To begin let H be a finitely generated free abelian group. A linear form on H is simply a group homomorphism α : H → ℤ. The set of all such forms with the operation of pointwise addition of homomorphisms forms a free abelian group called the dual group of H and denoted by H^{∗}. A symmetric bilinear form on H is a bilinear map λ : H ×H → ℤ such that λ(v, w) = λ(w, v) for all v, w ∈ H. There are two ways one might combine a bilinear form λ : H × H → ℤ and a linear form α : H → ℤ. One may follow Wall and simply consider the pair (λ, α). A morphism of pairs (λ_{0}, α_{0}) (λ_{1}, α_{1}) is a homomorphism Θ : H_{0} → H_{1} such that α_{0} = α_{1}∘Θ and λ_{0} = λ_{1}∘(Θ×Θ). Another way to combine λ and α is to take the quadratic form associated to λ and add α to it.

Definition 2.7. A function κ : H → ℤ is called a quadratic function if there exists a bilinear form λ on H, a linear functional α ∈ H^{∗} such that

κ(v) = λ(v, v) + α(v)

for all v ∈ H.

Proposition 2.8. The form λ and dual element α are uniquely recovered from κ using the following relations:

\begin{matrix}κ(v + w) = κ(v) + κ(w) + 2λ(v, w) \\ α(v) = κ(v) − λ(v, v). \end{matrix}

Proof. The first relation shows that λ is determined by κ and the second shows how to recover α from κ given that λ is known.

When we wish to specify the components of a quadratic function we shall write κ = κ(H, λ, α) for a quadratic function with domain H, bilinear form λ and linear

3. QUADRATIC FUNCTIONS 21

form α. On occasions we shall drop any of H, λ or α from the list when they are understood or not relevant. Note that κ is linear in the case where λ is identically zero. When α = 0, the quadratic function κ is homogeneous, in which case κ is a quadratic form.

We shall see below that what distinguishes the quadratic functions which arise from handlebodies of dimension 8 or 16 is that they take on only even integer values. We therefore make the following

Definition 2.9. An element α ∈ H^{∗} is called characteristic for λ when κ(λ, α) takes on only even values. That is, λ(v, v) + α(v) is even for all v ∈ H. In this case we shall call κ(λ, α) a characteristic quadratic function.

Now let κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}) be quadratic functions on H_{0} and H_{1} respectively and let Θ : H_{0} → H_{1} be a homomorphism. We define Θ^{∗}κ_{1} by Θ^{∗}κ_{1}(v) = κ_{1}(Θ(v)). A homomorphism Θ is called an isometric embedding if it is injective and Θ^{∗}κ_{1}(v) = κ_{0}(v) for every v ∈ H_{0}. A bijective isometric embedding is called an isometry. We observe that an isometry of quadratic functions induces an isometry of both the homogeneous quadratic and linear parts and record this fact as a proposition.

Proposition 2.10. Let λ_{0} and λ_{1} be symmetric bilinear forms on free abelian groups H_{0} and H_{1} respectively and let α_{0} and α_{1} be elements of H_{0}^{∗} and H_{1}^{∗} respectively. Then the the quadratic functions κ(H_{0}, λ_{0}, α_{0}) and κ(H_{1}, λ_{1}, α_{1}) are isometric if and only if the pairs (λ_{0}, α_{1}) and (λ_{1}, α_{1}) are isomorphic.

Proof. This follows immediately from Proposition 2.8.

Thus the two methods for combining λ and α which we mentioned earlier are equivalent.

Now let κ = κ(H, λ, α) be a quadratic function. The adjoint of the bilinear form λ is the linear transformation induced by λ from H to its dual,

\begin{matrix}λ̂ : H & → & H^{∗} \\ v & ↦ & [w ↦ λ(v, w)]. \end{matrix}

Thus for all v, w ∈ H, λ(v, w) = λ̂(v)(w). We define the radical of λ to be the group F := Ker(λ̂) = {v ∈ H|λ(v, w) = 0 ∀w ∈ H} and the quotient of λ to be the group G := Cok(λ̂) = H^{∗}/λ̂(H). The groups F, H, H^{∗} and G fit into the following exact sequence:

0 ⟶ F ⟶ H ⟶^{λ̂} H^{∗} ⟶^{π} G ⟶ 0

where π is the canonical projection from H^{∗} to H^{∗}/λ̂(H) = G. We call this sequence the fundamental sequence of λ and κ as it will occupy us for most of the remainder of this thesis. If F = 0 then λ and κ are called nondegenerate. If in addition G = 0 then λ and κ are called nonsingular.

Given two quadratic functions κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}), the direct sum κ_{0} ⊕ κ_{1} is defined on H_{0} ⊕ H_{1} via the equation

(κ_{0} ⊕ κ_{1})[(v_{0}, v_{1})] = κ_{0}(v_{0}) + κ_{1}(v_{1}).

We conclude this section by making the relation between handlebodies and quadratic functions explicit.

Definition 2.11. Let L ∈ H(4j) be a handlebody with intersection form λ(L) and tangential invariant α(L) defined on H_{4j}(L), then κ(L) := κ(H_{4j}(L), λ(L), α(L)) shall be called the quadratic function of L. Conversely, if κ = κ(H, λ, α) is a quadratic function such that there is an 8j-dimensional handlebody L with κ(L) ≅ κ(H, λ, α) then Ł^{8j}(κ) shall denote the 8j-dimensional handlebody defined up to diffeomorphism by κ.

4. WALL’S CLASSIFICATION RESTATED 23

Now let P denote the boundary of L, let P D : H_{4j}(L, P) ≅ H^{4j}(L) denote the Poincaré duality isomorphism and let j : H_{4j}(L) → H_{4j}(L, P) denote the map induced by the map of pairs (L, ϕ) → (L, P). The following fact is fundamental and well known. If we use the Kronecker pairing to identify H^{4j}(L) with Hom(H_{4j}(L), ℤ) then P D ∘j : H_{4j}(L) → H^{4j}(L) is the adjoint of the intersection pairing on L. Thus, given Definition 2.11 we have the following pair of isomorphic sequences

[figure: 0; H_{4j}(P); H_{4j}(L); H^{4j}(L); H^{4j}(P); 0; 0; F; H; H^{∗}; G; 0; i_{∗}; P D∘j; i^{∗}; λ̂; π]

where i_{∗} and i^{∗} denote the maps induced by the inclusion i : P ↪ L. This gives a topological interpretation of the fundamental sequence of κ(L) which we shall use throughout this thesis.

It is apparent that the boundary connected sum of handlebodies gives rise to the direct sum of the associated quadratic functions so that κ(L_{0}♮L_{1}) = κ(L_{0}) ⊕ κ(L_{1}). If we reverse the orientation on L then this will change the intersection form of L by reversing its sign. However, reversing the orientation on L does not affect the tangential invariant. This is because α(L) is the primary obstruction to trivializing the tangent bundle and we noted above that this class is a multiple of the Pontryagin class which is not affected by a change of orientation. Thus if κ = κ(H, λ, α) we define κ^{−} to be the quadratic function κ(H, −λ, α) and with this definition κ(−L) = κ(L)^{−}.

4. Wall’s classification restated

We may now restate Wall’s theorem 2.4 as follows: for every isometry Θ : κ(L_{0}) ≅ κ(L_{1}) there corresponds a diffeomorphism f(Θ) whose induced map on homology is Θ. We shall now dress this statement up in categorical terms by defining the following categories. Let H^{8j} := H(4j) be the category whose objects are highly connected 8j-manifolds L and whose morphisms are equivalence classes of diffeomorphisms of

4. WALL’S CLASSIFICATION RESTATED 24

handlebodies. Two diffeomorphisms g, h : L_{0} → L_{1} are equivalent if they induce the same isomorphism on middle homology, g_{∗} = h_{∗} : H_{4j}(L_{0}) → H_{4j}(L_{1}). We define F to be the category whose objects are quadratic functions on free abelian groups and whose morphisms are isometries of quadratic functions. Finally we define F^{8j} to be the full sub-category of F whose objects are quadratic functions which are isomorphic to the quadratic function of some handlebody L. A symmetric monoidal category is one with an associative and commutative addition operation on its objects that is functorially well behaved (see [MacL] VII 7). We shall not distract the reader with the precise definitions but assure him that H^{8j} and F^{∗} are symmetric monoidal categories with respect to the operations of boundary connected sum of handlebodies and of direct sum of quadratic functions.

Theorem 2.12 (Restatement of [Wa1] Theorem 1). There is an equivalence of (symmetric monoidal) categories

\begin{aligned}κ^{j} : & H^{8j} & ≅ & F^{8j} \\ & \{L, [g], ♮\} & → & \{κ(L), g_{∗}, ⊕\}. \end{aligned}

Proof. Recall that a skeleton of a category C is a full subcategory of C which contains precisely one object for every isomorphic class of objects in C. Every category C is equivalent to every one of its skeletons ([MacL] IV 4). So let \underline{H^{8j}} be a skeleton of H^{8j}. By Corollary 2.5, κ^{j}(\underline{H^{8j}}) is a skeleton of F^{8j} and therefore that κ is an equivalence of categories. Finally, for any pair of handlebodies L_{0} and L_{1} and any pair of diffeomorphisms f_{0} and f_{1}, κ(L_{0}♮L_{1}) = κ(L_{0}) ⊕ κ(L_{1}) and (f_{0}♮f_{1})_{∗} = f_{0∗} ⊕ f_{1∗}.

We now come to a fundamental observation which Wall and Wilkens did not fully exploit.

Proposition 2.13. When j ≠ 1, 2 the category F^{8j} has as objects all quadratic functions κ(λ, α) with even bilinear forms λ (and no restriction on α). When j = 1, 2 the category F^{8j} has objects all characteristic quadratic functions (with no restriction on the type of the bilinear form).

Proof. We have seen that a handlebody L is completely described by a presentation {(v_{1}, …, v_{l}), {c_{ij}}, ᾱ(1), …, ᾱ(l)} consisting of a basis of handles, linking numbers between the handles and the normal bundle of each handle. With respect to this basis the intersection matrix of L has components λ_{ij} where

λ_{ij} = \{ \begin{matrix}c_{ij} & i ≠ j \\ E(α(i)) & i = j \end{matrix} \}

and where E(α(i)) is the Euler number of ᾱ(i) ∈ π_{4j−1}(SO((4j)). When j ≠ 1, 2 Adams’ result on the non-existence of Hopf invariant one bundles ensures that each E(ᾱ(i)) is even and hence that λ is even. However, since every set of linking numbers may be realized by a link of (4j − 1)-spheres in S^{4j−1} we see that any even form may be produced. The stable tangential invariant is determined by the stabilizations S(ᾱ(i)) ∈ π_{4j−1}(SO) which are the values of α on the basis elements v_{i}. That these can be chosen arbitrarily follows from the result (see [Wa1, 171]) that the image of the homomorphism

S ⊕ E : π_{4j−1}(SO(4j)) → π_{4j−1}(SO) ⊕ ℤ

has index 2. Since E takes only even values for j ≠ 1, 2 the map S must be onto π_{4j−1}(SO). When j = 1, 2 then there are bundles with every possible Euler number and thus any intersection form one desires may be produced. Moreover, for every ᾱ ∈ π_{4j−1}(SO(4j)), we now have that S(ᾱ) and E(ᾱ) are of the same parity. Hence for i = 1, …, l,

λ(v_{i}, v_{i}) = E(ᾱ(i)) ≡ S(ᾱ(i)) (mod 2) ≡ α(v_{i}) (mod 2).

As {v_{1}, …, v_{l}} is a basis for H_{4j}(L) it follows that λ(v, v) ≡ α(v) mod (2) for every v ∈ H_{4j}(L).

4. WALL’S CLASSIFICATION RESTATED 26

As a consequence of Proposition 2.13 we define three categories of quadratic functions as follows.

• F^{c} shall denote the full sub-category of F whose objects are characteristic quadratic functions with arbitrary bilinear forms.

• F^{ev} shall denote the full sub-category of F whose objects are quadratic functions with even bilinear forms and arbitrary linear terms.

• F^{o} shall denote the full sub-category of F whose objects are even quadratic

forms. Proposition 2.13 states that F^{8j} = F^{c} when j = 1, 2 and F^{8j} = F^{ev} when j > 2. For all values of j the category F^{o} corresponds to the category of stably parallelizable handlebodies. Within each category F^{∗} there are full subcategories of nondegenerate and nonsingular quadratic functions. They correspond respectively to the categories of handlebodies cobounding rational homology spheres and handlebodies cobounding homotopy spheres. As we are primarily concerned with the classification of the boundaries of handlebodies up to connected sum with homotopy spheres we define the notion of stable F^{∗}-equivalence of quadratic functions.

Definition 2.14. We call two quadratic functions κ_{0} and κ_{1} belonging to the category F^{∗}, stably F^{∗}-equivalent if there are nonsingular quadratic functions μ_{0} and μ_{1} also belonging to F^{∗} and an isometry

Θ : κ_{0} ⊕ μ_{0} ≅ κ_{1} ⊕ μ_{1}.

We shall denote the stable F^{∗}-equivalence class of κ ∈ F^{∗} by [κ]_{F^{∗}}. Note the direct sum of stable F^{∗}-equivalence classes of quadratic functions is well defined,

[κ_{0}]_{F^{∗}} ⊕ [κ_{1}]_{F^{∗}} := [κ_{0} ⊕ κ_{1}]_{F^{∗}}.

5. HIGHLY CONNECTED BORDISM 27

5. Highly connected bordism

Our perspective on highly connected (8j − 1)-manifolds is to view them as the boundaries of handlebodies. However, not every highly connected (8j −1)-manifold P bounds a handlebody. In this section we digress somewhat to introduce an invariant of P which is the obstruction to P bounding a handlebody. With this invariant we shall show that every P bounds a handlebody after the addition of the right homotopy sphere. We remind the reader that an n-manifold is highly connected if all of its homotopy groups vanish below its middle dimension which is [n/2]. Now let Ω_{n}^{HC} be the set of highly connected bordism classes of highly connected, closed (smooth, oriented) manifolds. A highly connected bordism between highly connected n-manifolds M_{0} and M_{1} is a compact highly connected (n + 1)-manifold W and a diffeomorphism ∂W ≅ M_{0} ⊔ −M_{1}. The relation of being bordant via a highly connected manifold defines an equivalence relation. We shall write [M] for the highly connected bordism class of a highly connected manifold M and 0 for the class of the empty manifold. Note that [P] = 0 if and only if P is the boundary of a handlebody L.

Lemma 2.15. When n is even or when n = 2k−1 is odd and k ≡ 0, 4, 6, 7 (mod 8) the operations of connected sum and reversing orientation give rise to well defined operations on Ω_{n}^{HC} under which Ω_{n}^{HC} is an abelian group.

Proof. Given highly connected bordism classes [M_{0}] and [M_{1}] we define

[M_{0}] + [M_{1}] := [M_{0}#M_{1}] and − [M_{0}] := [−M_{0}].

We now verify that these operations are well defined and that [M# − M] = 0. If W is a highly connected bordism from M_{0} to M′_{0} then −W is a bordism from −M_{0} to −M′_{0} and so reversing orientation is well defined on highly connected bordism. Now, M#−M = ∂((M −D^{n})×[0, 1]) and if the dimension of M is even, (M −D^{n})×[0, 1] is

5. HIGHLY CONNECTED BORDISM 28

a highly connected zero bordism of M# − M. When n = 2k − 1 is odd the additional hypothesis on k entails that π_{k−2}(SO) = 0. This ensures that every stable bundle over S^{k−1} is trivial and hence that (M −D^{n})×[0, 1] is (k−1)-parallelizable in the language of [Mi]. We may therefore perform surgeries in the interior of (M − D^{n}) × [0, 1] to kill π_{k−1}. The resulting manifold is a highly connected zero bordism of M# − M.

Suppose that W_{0} is a highly connected bordism from M_{0} to M′_{0}. To show that connected sum is well defined over highly connected bordism we must show that there is a highly connected bordism from M_{0}#M_{1} to M′_{0}#M_{1}. We first note that there is a highly connected bordism W_{1} from M_{1} to −M_{1}. When n is even we take M_{1} × [0, 1] and when n = 2k − 1 we argue as before that surgery may be performed in the interior of M_{1} × [0, 1] to produce a highly connected bordism. The band connected sum of W_{0} and W_{1} (see [KM] §2) is then a highly connected bordism from M_{0}#M_{1} to M′_{0}#M_{1}. It is clear that the connected sum operation on highly connected bordism is associative and commutative. It follows that Ω_{n}^{HC} is an abelian group in the indicated dimensions.

Lemma 2.16. Let k > 1 and k ≡ 0, 4, 6, 7 (mod 8). Then every member of Ω_{2k−1}^{HC} is represented by a homotopy sphere and

Ω_{2k−1}^{HC} ≅ Θ_{2k−1}/bP_{2k}.

Proof. We let

Ω : Θ_{2k−1} → Ω_{2k−1}^{HC}

be the homomorphism we sends a homotopy sphere to the highly connected bordism class which it represents. It follows from calculations of Wall in [Wa1] that the kernel of Ω is precisely bP_{2k}. It remains to show that Ω is onto but this is a simple consequence of the success of simply connected surgery in odd dimensions as we now

5. HIGHLY CONNECTED BORDISM 29

show. One may always take the (k −1)-skeleton of a closed highly connected (2k −1)- manifold M to be a wedge of (k − 1)-spheres. From the assumption on k it follows that that π_{k−2}(SO) = 0 and thus every stable bundle over a wedge of (k − 1)-spheres is trivial. In particular the tangent bundle of M restricted to the (k − 1)-skeleton of M is trivial. Thus M is (k − 1)-parallelizable and we may perform a series of surgeries on (k − 1) spheres in M to kill π_{k−1}(M). The resulting (2k − 1)-manifold is a homotopy sphere Σ and the trace of the surgeries performed is a highly connected bordism from M to Σ.

Corollary 2.17. Let P be a highly connected (8j − 1)-manifold. There is a unique homotopy sphere Σ_{P} such that P #Σ_{P} is the boundary of a handlebody and s(Σ_{P}) = 0 ∈ bP_{8j−1}. Moreover, if P_{0} and P_{1} are two highly connected (8j − 1)- manifolds then Σ_{P_{0}#P_{1}} = Σ_{P_{0}}#Σ_{P_{1}}.

Proof. We note first that 8j − 1 is a dimension to which both Lemmas 2.15 and 2.16 apply. By Lemma 2.16, −P is highly connected bordant to a homotopy sphere Σ which is determined up to connected sum with homotopy spheres in bP_{8j}. There is thus a unique homotopy sphere Σ_{P} such that s(Σ_{P}) = 0 and [Σ_{P}] = −[P] ∈ Ω_{8j−1}^{HC}. Now by Lemma 2.15, [P #Σ_{P}] = 0 and therefore P #Σ_{P} bounds a handlebody. To prove the second assertion we need only note that if L_{i} are handlebodies such that P_{i}#Σ_{P_{i}} = ∂L_{i} for i = 0, 1 then

(P_{0}#P_{1})#(Σ_{P_{0}}#Σ_{P_{1}}) = ∂(L_{0}♮L_{1}) and s(Σ_{P_{0}}#Σ_{P_{1}}) = s(Σ_{P_{0}}) + s(Σ_{P_{1}}) = 0.

We remind the reader that in the dimensions of primary concern in this thesis Θ_{7}/bP_{8} = 0 and Θ_{15}/bP_{16} = ℤ/2. In particular, in dimension 7, the homotopy sphere Σ_{P} is always the standard sphere and every manifold P bounds a handlebody.

6. DECOMPOSITION OF QUADRATIC FUNCTIONS 30

6. Decomposition of quadratic functions

Quadratic functions are of two essentially different types. There are linear or “totally degenerate” quadratic functions at one extreme and at the other there are nondegenerate quadratic functions. In this section we consider the decomposition of a general quadratic function into the sum of linear and nondegenerate quadratic functions.

Let κ = κ(H, λ, α) be a quadratic function and recall the fundamental sequence of κ,

0 ⟶ F ⟶ H ⟶^{λ̂} H^{∗} ⟶^{π} G ⟶ 0.

By definition F is the subspace of H on which κ is linear. It is clear that F is a summand of H for suppose that r.v ∈ F for some non-zero integer r. This means that λ(rv, w) = rλ(v, w) = 0 for every w ∈ H. Since r is non-zero we conclude that λ(v, w) = 0 for every w ∈ H and that v belongs to F. Thus F is a summand of H and H/F is a free group. The bilinear form λ descends to define a nondegenerate bilinear form on H/F in the familiar way:

\begin{aligned}λ_{/F} : & H/F × H/F & → & ℤ \\ & (v + F, w + F) & ↦ & λ(v, w). \end{aligned}

The function λ_{/F} is well defined because different coset representatives differ by an element of F but λ(u, v) = 0 for any u ∈ F and any v ∈ H. Moreover, if λ/F (v + F, w + F) = 0 for every w + F ∈ H/F then v ∈ F, v + F = 0 and so λ_{/F} is nondegenerate. Now let ρ : H → H/F denote the canonical projection from H onto H/F and let

Sec(ρ) := {Ψ : H/F → H|ρ ∘ s = Id_{H/F} }

6. DECOMPOSITION OF QUADRATIC FUNCTIONS 31

denote the set of sections of ρ. The set Sec(ρ) is nonempty and there is a free and transitive action of Hom(H/F, F) on Sec(ρ) defined by

(ψ · Ψ)(v + F) = Ψ(v + F) + ψ(v + F)

where ψ ∈ Hom(H/F, F), Ψ ∈ Sec(ρ) and v + F ∈ H/F. For every Ψ ∈ Sec(ρ), we define the projections

\begin{aligned}p_{F} (Ψ) : & H & → & F \\ & v & ↦ & v − Ψ(ρ(v)) \end{aligned}  and  \begin{aligned}p(Ψ) : & H & → & Ψ(H/F) \\ & v & ↦ & Ψ(ρ(v)) \end{aligned}

and the subgroups

H(Ψ) := Ψ(H/F) H^{∗}(Ψ) := p(Ψ)^{∗}(H(Ψ)^{∗})
F^{∗}(Ψ) := p_{F} (Ψ)^{∗}(F^{∗}) G(Ψ) = π(F^{∗}(Ψ)) ⊂ G.

It is straightforward to check that there are direct sum decompositions H = F ⊕H(Ψ), H^{∗} = F^{∗}(Ψ) ⊕ H^{∗}(Ψ) and G = G(Ψ) ⊕ T G where T G denotes the torsion subgroup of G. Moreover, we show below that H^{∗}(Ψ) ⊂ H^{∗} does not depend upon Ψ and so we shall henceforth identify H(Ψ)^{∗} and H^{∗}(Ψ) via p(Ψ)^{∗}. The reason for doing this is that it is technically convenient to think of H(Ψ)^{∗} as a subgroup of H^{∗}. We denote the restrictions of α, λ and κ as follows

α(Ψ) := α|_{H(Ψ)}, λ(Ψ) := λ|_{H(Ψ)×H(Ψ)} and κ(Ψ) := κ|_{H(Ψ)}.

With this notation κ(Ψ) = κ(H(Ψ), λ(Ψ), α(Ψ)), α = α(Ψ) + p_{F} (Ψ)^{∗}(α|_{F}). We also obtain the following decomposition of the fundamental sequence of κ:

0 ⟶ F ⟶^{=} F ⟶^{0} F^{∗}(Ψ) ⟶ G(Ψ) ⟶ 0
⨁ ⨁ ⨁
0 ⟶ H(Ψ) ⟶^{λ̂(Ψ)} H^{∗}(Ψ) ⟶ T G ⟶ 0

where λ̂(Ψ) denotes the adjoint of λ(Ψ).

Proposition 2.18. Let κ(H, λ, α) be a quadratic function and λ_{/F}, ρ and Sec(ρ) be as above. Then, for every section Ψ ∈ Sec(ρ) and for every ψ ∈ Hom(H/F, F),

6. DECOMPOSITION OF QUADRATIC FUNCTIONS 32

1. Ψ^{∗}(λ) = λ_{/F},

2. Ψ and ρ|_{H(Ψ)} define inverse isometries Ψ : Ψ^{∗}(κ) ≅ κ(Ψ) and ρ|_{H(Ψ)} : κ(Ψ) ≅ Ψ^{∗}(κ),

3. κ(Ψ) is nondegenerate and κ = κ(Ψ) ⊕ α|_{F},

4. H^{∗}(Ψ) = ρ^{∗}((H/F)^{∗}) and T G = H^{∗}(Ψ)/λ̂(Ψ)(H(Ψ)),

5. α(ψ · Ψ) = α(Ψ) + ρ^{∗}(ψ^{∗}(α|_{F})).

Proof. (1.) Let v, w ∈ H be arbitrary and let v′ = Ψ(v +F) and w′ = Ψ(w +F) so that v′ + F = v + F and w′ + F = w + F. We calculate that

Ψ^{∗}(λ)(v + F, w + F) = λ(Ψ(v + F), Ψ(w + F))
= λ(v′, w′)
= λ_{/F} (v + F, w + F).

(2.) This is true by definition. Certainly Ψ ∘ ρ|_{H(Ψ)} = Id_{H(Ψ)} and ρ|_{H(Ψ)} ∘ Ψ = Id_{H/F}. Moreover, for v ∈ H(Ψ) we calculate that

\begin{aligned}κ(Ψ)(Ψ(v + F)) & = κ(Ψ(v + F)) \\ & = Ψ^{∗}(κ)(v + F) \end{aligned} and \begin{aligned}Ψ^{∗}(κ)(ρ|_{H(Ψ)}(v)) & = κ(Ψ(ρ|_{H(Ψ)}(v)) \\ & = κ(v) \\ & = κ(Ψ)(v). \end{aligned}

(3.) We showed above the that λ_{/F} is nondegenerate and so by 1 we see that Ψ^{∗}(κ) is nondegenerate and then by 2 we have that κ(Ψ) is nondegenerate. For each v ∈ H, we decompose v as v = v − Ψ(ρ(v)) + Ψ(ρ(v)) where v − Ψ(ρ(v)) ∈ F and Ψ(ρ(v)) ∈ H(Ψ). Therefore

κ(v) = λ(Ψ(ρ(v)), Ψ(ρ(v))) + α(Ψ(ρ(v))) + α(v − Ψ(ρ(v)))
= κ(Ψ)(Ψ(ρ(v)) + α|_{F} (v − Ψ(ρ(v)))
= (κ(Ψ) ⊕ α|_{F})[(Ψ(ρ(v)), v − Ψ(ρ(v)))].

(4.) Since this result justifies our convention identifying H^{∗}(Ψ) and H(Ψ)^{∗} we briefly drop that convention. For every Ψ, ρ^{∗} : (H/F)^{∗} → H^{∗} factors as

(H/F)^{∗} ⟶^{(ρ|_{H(Ψ)})^{∗}} H(Ψ)^{∗} ⟶^{p(Ψ)^{∗}} H^{∗}(Ψ) ⊂ H^{∗}

7. LINKING FORMS AND KRONECKER PAIRINGS 33

as the following calculation demonstrates. For any z ∈ (H/F)^{∗} and any v ∈ H

p(Ψ)^{∗} ∘ (ρ|_{H(Ψ)})^{∗}(z)(v) = (ρ|_{H(Ψ)})^{∗}(z)((Ψ ∘ ρ)(v))
= z((ρ|_{H(Ψ)} ∘ Ψ ∘ ρ)(v))
= z(ρ(v))
= ρ^{∗}(z)(v).

As (ρ|_{H(Ψ)})^{∗} and p(Ψ)^{∗} are isomorphisms it follows that H^{∗}(Ψ) = p(Ψ)^{∗}(H(Ψ)^{∗}) = ρ^{∗}((H/F)^{∗}). The second assertion follows from the splitting of the fundamental sequence of κ just prior to the statement of this proposition. Equivalently it may demonstrated as follows. For every v ∈ H, λ̂(v) = λ̂(Ψ(ρ(v))) and for every w ∈ H,

\begin{aligned}λ(v, w) & = λ(v − Ψ(ρ(v)) + Ψ(ρ(v)), w) \\ & = λ(Ψ(ρ(v)), w). \end{aligned}

It follows that λ̂(H) = λ̂(Ψ)(H(Ψ)) and that

Cok(λ̂) = F^{∗}(Ψ) ⊕ H^{∗}(Ψ)/λ̂(Ψ)(H(Ψ)) = F^{∗}(Ψ) ⊕ Cok(λ̂(Ψ)).

Moreover, as λ(Ψ) is nondegenerate, Cok(λ̂(Ψ)) is a torsion group and thus it is the torsion subgroup of Cok(λ̂). (5.) Given v ∈ H, α(Ψ)(v) = α(Ψ(ρ(v))) and thus

α(ψ · Ψ)(v) = α(Ψ(ρ(v)) + ψ(ρ(v)))
= α(Ψ(ρ(v))) + α|_{F} (ψ(ρ(v)))
= α(Ψ)(v) + ρ^{∗}(ψ^{∗}(α|_{F}))(v).

7. Linking forms and Kronecker pairings

In this section we review the algebraic structures that a quadratic function κ = κ(H, λ, α) induces on its quotient G and radical F. The most important of these is the linking form induced on the torsion subgroup of G. By a linking form on a finite

7. LINKING FORMS AND KRONECKER PAIRINGS 34

abelian group T G we shall mean a function

b : T G × T G → ℚ/ℤ

with the following properties.

1. Symmetry: b(x, y) = b(y, x).

2. Bilinearity: b(x, y + z) = b(x, y) + b(x, z).

3. Nonsingularity: b(x, y) = 0 ∀x ∈ T G ⇒ y = 0. In Chapter 5 we shall describe in detail the classification of linking forms on finite abelian groups but for now we report some fundamental facts concerning linking forms. If (T G_{0}, b_{0}) and (T G_{1}, b_{1}) are linking forms then an isomorphism θ : T G_{0} → T G_{1} is an isometry from b_{0} to b_{1} if b_{0} = b_{1} ∘ (θ × θ). There is a linking form (T G_{0} ⊕ T G_{1}, b_{0} ⊕ b_{1}) defined by

(b_{0} ⊕ b_{1})[(x_{0}, x_{1}), (y_{0}, y_{1})] = b_{0}(x_{0}, y_{0}) + b_{1}(x_{1}, y_{1}).

If (T G, b) is a linking form and T G_{0} is a subgroup of T G then the annihilator of T G_{0} is the set

T G_{0}^{⊥} := {x ∈ T G : b(x, y) = 0 ∀y ∈ T G_{0}}.

Lemma 2.19. Let (T G, b) be a linking form. If T G_{0} is a finite subgroup of T G such that b|_{T G_{0}×T G_{0}} : T G_{0} × T G_{0} → ℚ/ℤ is nonsingular then b|_{T G_{0}^{⊥}×T G_{0}^{⊥}} is nonsingular and

b ≅ b|_{T G_{0}⊕T G_{0}} ⊕ b|_{T G_{0}^{⊥}×T G_{0}^{⊥}}.

Proof. We postpone the proof which is straight forward until chapter 5. A linking form (T G, b) is called indecomposable if it cannot be written as a non-trivial direct sum of sublinking forms. In [Wa2] Wall proved that the groups which arise as the domains of indecomposable linking forms are ℤ/p^{k} and ℤ/2^{k} ⊕ ℤ/2^{k} for p any prime and k any positive integer.

7. LINKING FORMS AND KRONECKER PAIRINGS 35

Every quadratic function κ(H, λ, α) defines a linking form on the torsion subgroup of its quotient group T G ⊂ G = Cok(λ̂) via the bilinear form λ. We begin by assuming that λ is nondegenerate in which case the fundamental sequence of λ is

0 ⟶ H ⟶^{λ̂} H^{∗} ⟶^{π} T G ⟶ 0.

If we tensor this sequence with ℚ and λ̂ with Id_{ℚ} then λ̂ ⊗ Id_{ℚ} is an isomorphism. We let λ̂^{−1} denote the restriction of the inverse of λ̂ ⊗ Id_{ℚ} to H^{∗},

λ̂^{−1} := (λ̂ ⊗ Id_{ℚ})^{−1}|_{H^{∗}} : H^{∗} → H ⊗ ℚ.

The linear map λ̂^{−1} : H^{∗} → H ⊗ ℚ defines a bilinear pairing

\begin{aligned}λ^{−1} : & H^{∗} × H^{∗} & → & ℚ \\ & (x, y) & ↦ & y(λ̂^{−1}(x)) \end{aligned}

where y(v⊗ \frac{p}{q}) = \frac{p}{q}y(v). The symmetry of λ̂^{−1} follows immediately from the symmetry of λ. The pairing λ̂^{−1} may be defined alternatively as follows. For any x, y ∈ H^{∗} there is a non-zero r ∈ ℤ and some v ∈ H such that rx = λ̂(v). We have the following identity since λ̂^{−1}(x) = \frac{1}{r}v,

λ^{−1}(x, y) = \frac{1}{r}y(v).

The bilinear pairing λ^{−1} induces a nonsingular linking form on T G as follows. Let [x] = π(x) and [y] = π(y) ∈ T G.

\begin{aligned}b = b(λ) : & T G × T G & → & ℚ/ℤ \\ & ([x], [y]) & ↦ & λ^{−1}(x, y) & mod ℤ \end{aligned}

The linking form b is well defined in ℚ/ℤ, for suppose that x + λ̂(v) and y + λ̂(w) are different lifts of [x] and [y]. Then

λ^{−1}(x + λ̂(v), y + λ̂(w)) = λ^{−1}(x, y) + λ^{−1}(x, λ̂(w)) + λ^{−1}(λ̂(v), y)
+λ^{−1}(λ̂(v), λ̂(w)) mod ℤ
= λ^{−1}(x, y) + x(w) + y(v) + λ(v, w) mod ℤ
= λ^{−1}(x, y) mod ℤ.

7. LINKING FORMS AND KRONECKER PAIRINGS 36

The function b(λ) is evidently bilinear and symmetric. It is also nonsingular as we now show. We may choose an indivisible v ∈ H and an integer r such that rx = λ̂(v) for if v′ ∈ H and r′ ∈ ℤ satisfy x = r′λ̂(v′) then if v′ = s.v for s ∈ ℤ and v indivisible then x = r′sλ̂(v). Now, if If b([x], [y]) = 0 for every [y] ∈ T G then \frac{1}{r}y(v) ∈ ℤ for every y ∈ H^{∗}. Since v is indivisible there is a y ∈ H^{∗} such that y(v) = 1. Thus r = ±1 and hence [x] = 0 ∈ T G. We may also define b by its adjoint map to the torsion dual of T G, T G^{∧} = Hom(T G, ℚ/ℤ),

\begin{aligned}b̂: & T G & → & T G^{∧} \\ & [x] & ↦ & ([y] ↦ b([x], [y])). \end{aligned}

As T G and T G^{∧} are finite groups of the same order b̂ is an isomorphism if and only if b is nonsingular.

If the quadratic function κ is degenerate then by Proposition 2.18, each section Ψ defines a nondegenerate quadratic function κ(Ψ) where T G = H^{∗}(Ψ)/λ̂(Ψ)(H(Ψ)) and H^{∗}(Ψ) = ρ^{∗}((H/F)^{∗}) are is independent of Ψ. In fact the following is true.

Lemma 2.20. The bilinear pairing λ(Ψ)^{−1} : H^{∗}(Ψ) × H^{∗}(Ψ) → ℚ is independent of Ψ.

Proof. Suppose that Ψ′ is another section of ρ, that x, y ∈ H^{∗}(Ψ′) = H^{∗}(Ψ) and that rx = λ̂(Ψ)(v) for v ∈ H(Ψ) and a non-zero integer r. If we set v′ := Ψ′(ρ(v)), then λ̂(Ψ′)(v) = rx, v′ − v ∈ F and hence y(v′ − v) = 0. We calculate that

λ^{−1}(Ψ)(x, y) = \frac{1}{r}y(v)
= \frac{1}{r}y(v + v′ − v)
= \frac{1}{r}y(v′)
= λ^{−1}(Ψ′)(x, y).

7. LINKING FORMS AND KRONECKER PAIRINGS 37

It follows from Lemma 2.20 that b(λ(Ψ)) = b(λ(Ψ′)) and so given a degenerate bilinear form λ, we define

b(λ) := b(λ(Ψ))

for any section Ψ.

In addition to the linking form b(λ), a bilinear form λ also defines a Kronecker pairing between F = Ker(λ̂) and G = Cok(λ̂) by

\begin{aligned}λ̄ : & F × G & → & ℤ \\ & (v, [x]) & ↦ & x(v). \end{aligned}

The pairing λ̄ is well defined because any two choices of lifts of [x] differ by λ̂(w) for some w ∈ H and λ̂(w)(v) = 0. We label the adjoint homomorphisms associated to λ̄ by λ̄_{F} : F → G^{∗} and λ̄_{G} : G → F^{∗} where λ̄_{F} (v)([x]) = λ̄(v, [x]) and λ̄_{G}([x])(v) = λ̄(v, [x]). Here we have extended the ^{∗} notation to any abelian group G, so that G^{∗} := Hom(G, ℤ) denotes the group of homomorphism from G to ℤ. Recall the canonical homomorphism

\begin{aligned}τ_{G} : & G & → & G^{∗∗} \\ & g & ↦ & (x ↦ x(g)). \end{aligned}

Lemma 2.21. The composition λ̄_{F}^{∗} ∘ τ_{G} : G → F^{∗} is equal to λ̄_{G}. The homomorphisms and λ̄_{G} : G → F^{∗} and λ̄_{F} : F → G^{∗} are respectively surjective and bijective.

Proof. The first assertion is a simple matter to check. Let [x] ∈ G and v ∈ F,

λ̄_{F}^{∗} (τ_{G}([x]))(v) = τ_{G}([x])(λ̄_{F} (v))
= λ̄_{F} (v)([x])
= x(v)
= λ̄_{G}([x])(v).

In Section 6 we noted that choosing a section Ψ ∈ Sec(ρ) defines a splitting H^{∗} = F^{∗}(Ψ) ⊕ H^{∗}(Ψ) and the inclusion p_{F} (Ψ)^{∗} : F^{∗} ≅ F^{∗}(Ψ) ↪ H^{∗}. Given x ∈ F^{∗} it is

7. LINKING FORMS AND KRONECKER PAIRINGS 38

a matter of definitions that λ̄_{G}(π(p_{F} (Ψ)^{∗}(x))) = x and therefore λ̄_{G} is onto. Since λ̄_{G} = λ̄_{F}^{∗} ∘ τ_{G}, it follows that λ̄_{F}^{∗} in onto. Now note that F and G have the same rank by the Rank-Nullity Theorem applied to λ̂. Thus G^{∗∗} and F^{∗} have the same rank and so λ̄_{F}^{∗} is a surjective homomorphism between free abelian groups of the same rank and thus an isomorphism. This entails that λ̄_{F} is itself an isomorphism.

We now review and relate the linking form and Kronecker pairing of a bilinear form from a slightly more abstract perspective. Let A and B be free abelian groups and f : A → B a homomorphism with kernel F and cokernel G. Again we let ρ : A → A/F denote the canonical projection. Then there is a map b̂(f) : ρ^{∗}((A/F)^{∗}) → T G^{∧} constructed in a similar manner as before. If [b] ∈ T G then [b] = π(b) for some b ∈ B and rb = f(a) for some non-zero integer r and some a ∈ A. For x ∈ ρ^{∗}((A/F)^{∗}), x(a) = 0 for every a ∈ F, and so we define may

\begin{aligned}b̂(f) : & ρ^{∗}((A/F)^{∗}) & → & T G^{∧} \\ & x & ↦ & ([b] ↦ \frac{1}{r}x(a) ∈ ℚ/ℤ). \end{aligned}

Now choose any section Ψ : A/F → A so that we have the decomposition A^{∗} = ρ^{∗}((A/F)^{∗}) ⊕ F^{∗}(Ψ). With respect to this splitting define π̂ : A^{∗} → F^{∗} ⊕ T G^{∧} by π̂(x, y) = (y|_{F} , b̂(x)). It is easy to check that we now have the following pair of exact sequences:

[figure: 0; F; A; B; G; 0; 0; G^{∗}; B^{∗}; A^{∗}; F^{∗} ⊕ T G^{∧}; 0; f; π; π^{∗}; f^{∗}; π̂]

Observe also that b̂(f) is identically zero on the image of f and thus defines a function which we also denote b̂(f) : (A/F)^{∗}/f(A) → T G^{∧}. If we return to the case where B = A^{∗} and f = λ̂ is the adjoint of a bilinear form on A then b̂(λ̂) is the adjoint map of the linking form b(λ) which we defined above. Setting G(Ψ) := π(F^{∗}(Ψ)) ⊂ G the above pair of sequences becomes linked into the following commutative diagram

7. LINKING FORMS AND KRONECKER PAIRINGS 39

where each vertical arrow denotes and isomorphism.

[figure: 0; F; A; A^{∗}; G(Ψ) ⊕ T G; 0; 0; G^{∗}; A^{∗∗}; A^{∗}; F^{∗} ⊕ T G^{∧}; 0; λ̄_{F}; λ̂; τ_{A}; π; Id; ( \begin{matrix}λ̄_{G}|_{G(Ψ)} & 0 \\ 0 & b̂(λ̂) \end{matrix} ); π^{∗}; λ̂^{∗}; π̂]

Recall that if a handlebody L has quadratic function κ = κ(H, λ) and boundary P then the topological correlates of F = Ker(λ̂) and G = Cok(λ̂) are H_{4j}(P) and H^{4j}(P) respectively. The pairing λ̄ corresponds to the Kronecker pairing H_{4j}(P)×H^{4j}(P) → ℤ and this gives an alternate explanation of Lemma 2.21. The linking form b(λ) is the linking form of P which is a defined on the torsion subgroup of H^{4j}(P). Recall also that for every manifold P there is a specific homotopy Σ_{P} such that P #Σ_{P} bounds a handlebody, L say. Now there is a canonical homeomorphism P → P #Σ_{P} and therefore a canonical identification of H^{4j}(P) with H^{4j}(P #Σ_{P}).

Definition 2.22. Let P be a highly connected (8j − 1)-manifold and L a handlebody cobounding P #Σ_{P} with quadratic function κ(L) = κ(λ). The linking form of P is defined to be the linking form induced by λ;

b(P) := b(λ).

Remark 2.23. The linking form of P has at least two other definitions: by way of Poincaré duality with ℚ/ℤ coefficients ([KM]) and by letting r times a cycle representing a homology class bound and computing intersection (mod r) ([Wa3]). We leave it as an exercise to verify the well known fact that all three definitions give the same linking form. That our definition of b(P) is independent of the choice of handlebody will follow from results of Wilkens’ thesis which we review in Section 8.

Now let λ_{0} and λ_{1} be a pair of bilinear forms. It is clear from the definitions that b(λ_{0} ⊕ λ_{1}) = b(λ_{0}) ⊕ b(λ_{1}). We claim for two highly connected manifolds P_{0} and P_{1}

8. WILKENS’ CLASSIFICATION OF THE MANIFOLDS P 40

that

b(P_{0}#P_{1}) = b(P_{0}) ⊕ b(P_{1}).

For firstly by Corollary 2.17, Σ_{P_{0}#P_{1}} = Σ_{P_{0}}#Σ_{P_{1}} and thus if ∂L_{i} = P_{i}#Σ_{P_{i}} for i = 0, 1, then (P_{0}#P_{1})#Σ_{P_{0}#P_{1}} = ∂(L_{0}♮L_{1}). But now λ(L_{0}♮L_{1}) = λ(L_{0}) ⊕ λ(L_{1}) and our claim follows.

8. Wilkens’ classification of the manifolds P

The fundamental topological result in Wilkens’ thesis is the following theorem (whose proof we shall present in Chapter 4).

Theorem 2.24 (Wilkens’ Thesis 3.2). Let L_{0} and L_{1} be 2k-dimensional handlebodies (k > 2) with boundaries ∂L_{0} = P_{0} and ∂L_{1} = P_{1}. Suppose that f : P_{0} → P_{1} is a diffeomorphism. Then there are handlebodies V_{0}, V_{1} with boundaries the standard sphere and a diffeomorphism g which makes the following diagram commute.

[figure: P_{0}; P_{1}; L_{0}♮V_{0}; L_{1}♮V_{1}; f; g]

In the light of Theorem 2.24 and recalling Definition 2.14 which defined the stable F^{∗}-equivalence class of a quadratic function, we make the following

Definition 2.25. Let P be a highly connected (8j − 1)-manifold and L a handlebody such that ∂L = P #Σ_{P}. We define the stable quadratic function of P, [κ(P)], to be the stable F^{∗}-equivalence class of κ(L).

Corollary 2.26 (To Wilkens’ 2.24 and Wall’s 2.12). The stable quadratic function of a highly connected manifold P, [κ(P)], is a well defined and complete almost diffeomorphism invariant of P.

8. WILKENS’ CLASSIFICATION OF THE MANIFOLDS P 41

Proof. We show that [κ(P)] is both well defined and invariant under almost diffeomorphisms at the same time. Let P_{0} and P_{1} be highly connected manifolds, Σ a homotopy sphere and f : P_{0} → P_{1}#Σ a diffeomorphism. Then Σ_{P_{0}} = Σ_{P_{1}#Σ} = Σ_{P_{1}}#Σ_{Σ} and there is a diffeomorphism

f′ : P_{0}#Σ_{P_{0}} → P_{1}#Σ#Σ_{P_{0}} = P_{1}#Σ_{P_{1}}#Σ_{Σ}#Σ.

Let L_{0}, L_{1} and L_{Σ} be handlebodies bounding respectively P_{0}#Σ_{P_{0}}, P_{1}#Σ_{P_{1}} and Σ#Σ_{Σ}. Applying Theorem 2.24 to f′, L_{0} and L_{1}♮L_{Σ} we deduce that there are handle bodies V_{0} and V_{1} bounding standard spheres and a diffeomorphism

g : L_{0}♮V_{0} → L_{1}♮L_{Σ}♮V_{1}

which extends f′. So by Wall’s classification of handlebodies there is an isometry

Θ(g) : κ(L_{0}) ⊕ κ(V_{0}) → κ(L_{1}) ⊕ κ(L_{Σ}) ⊕ κ(V_{1}).

But the boundaries of V_{0}, V_{1} and L_{Σ} are all homotopy spheres, so κ(V_{0}), κ(V_{1}) and κ(L_{Σ}) are all nonsingular quadratic functions in F^{∗} and thus Θ(g) defines a stable F^{∗}- equivalence from κ(L_{0}) to κ(L_{1}). That is, [κ(P_{0})] = [κ(L_{0})]_{F^{∗}} = [κ(L_{1})]_{F^{∗}} = [κ(P_{1})]. If we take P_{0} = P_{1} = P, Σ = S^{8j−1} and f = Id_{P} then this argument shows that [κ(P)] is independent of the handlebody chosen and hence well defined.

Finally we show that [κ(P)] is a complete invariant. Suppose that [κ(P_{0})] = [κ(P_{1})]. Then there are nonsingular quadratic functions μ_{0} and μ_{1} belonging to F^{∗} and an isometry

Θ : κ(L_{0}) ⊕ μ_{0} → κ(L_{1}) ⊕ μ_{1}.

Again by Wall’s classification of handlebodies there are handlebodies L(μ_{0}) and L(μ_{1}) which cobound homotopy spheres Σ_{0} and Σ_{1} and there is a diffeomorphism implementing Θ

h(Θ) : L_{0}♮L(μ_{0}) → L_{1}♮L(μ_{1}).

8. WILKENS’ CLASSIFICATION OF THE MANIFOLDS P 42

When restricted to the boundary of L_{0}♮L(μ_{0}), h(Θ) yields a diffeomorphism

h′ : P_{0}#Σ_{P_{0}}#Σ_{0} → P_{1}#Σ_{P_{1}}#Σ_{1}

and therefore P_{0} and P_{1} are almost diffeomorphic.

As Wilkens worked with triples (H, λ, α) as opposed to quadratic functions κ(H, λ, α) he would not have stated Corollary 2.26 as it is stated here. However, the equivalent statement for triples (H, λ, α) was known to Wilkens and was indeed central to his thesis. In the language of this thesis, the algebraic task which Wilkens undertook was to find further algebraic invariants of stable quadratic functions in F^{c} which would allow him to determine when two quadratic functions in F^{c} were stably equivalent. He was partially successful. Given a triple (H, λ, α) Wilkens extracted the triple (G, b, β). Where G = Cok(λ̂), b is the linking form induced on G and β is the image of α ∈ H^{∗} under the canonical map H^{∗} → H^{∗}/λ̂(H) = Cok(λ̂). The topological interpretation of this triple is by now largely familiar. We suppose that L is a 8 or 16 dimensional handlebody with quadratic function κ(H, λ, α) and with boundary P. Then G = H^{4j}(P), b = b(P) and if i : P ↪ L denotes the inclusion then β = i^{∗}(α) is the tangential invariant of P which is the primary obstruction to the stable triviality of the tangent bundle of P.

Wilkens’ first achievement was to show that the manifolds P split via connected sum according to the splitting of their linking forms. To be precise, Wilkens defined P to be decomposable if P can be expressed as a connected sum of manifolds neither of which is a homotopy sphere and he defined P to be indecomposable if is it not decomposable.

Proposition 2.27 (Wilkens Thesis). A manifold P is indecomposable if and only if either H^{4j}(P) ≅ ℤ or H^{4j}(P) is finite and the linking form of P is indecomposable. To every decomposition of H^{4j}(P) as a sum of infinite cyclic groups and finite groups over which the linking form of P is indecomposable there corresponds a decomposition of P by indecomposable manifolds.

Wilkens’ second achievement was to largely classify the indecomposable manifolds.

Theorem 2.28 (Wilkens’ Thesis). Let P be indecomposable and let P have invariants (G, b, β). If G contains no 2-torsion then the triple (G, b, β) is a complete almost diffeomorphism invariant of P. If G contains 2-torsion then there are at most two almost diffeomorphism classes of manifolds with the same invariants. A triple (G, b, β) with b indecomposable is ambiguous (i.e. is realized by a pair of not almost diffeomorphic manifolds) if and only if |G| ≤ 4 and β = 0 or if |G| > 4 and β is not divisible by 4.

9. Quadratic linking functions

In this section we define the quadratic linking functions which are the invariants required to settle the ambiguity in Wilkens classification. We also prove elementary lemmas concerning Q(b), the set of quadratic linking functions which refine a given linking form b, and then describe how a nondegenerate quadratic function induces a quadratic linking function on its quotient group.

Definition 2.29. A quadratic linking function q : T G → ℚ/ℤ on a finite abelian group T G is a function such that the function b(q) : T G × T G → ℚ/ℤ defined by

b(q)(x, y) := q(x + y) − q(x) − q(y) ∈ ℚ/ℤ

is a linking form on T G. We say that q is a quadratic refinement of b(q). If q satisfies q(x) = q(−x) for all x ∈ T G then we say that q is homogeneous and we call homogeneous quadratic linking functions quadratic linking forms.

Lemma 2.30. Let (T G, b) be a bilinear form. Then 1. there is a quadratic linking form q : T G → ℚ/ℤ which refines b,

2. if the order of G is odd then q is unique.

Proof. Choose a basis (minimal set of generators each of order p^{k}, p a prime and k a positive integer) {[x_{1}] …, [x_{n}]} for T G. If the order of [x_{i}] is a power of 2 then let q([x_{i}]) be either of the members of ℚ/ℤ such that 2q([x_{i}]) = b([x_{i}], [x_{i}]). If the order of [x_{i}] is odd choose the unique fraction with an odd denominator from amongst the two choices. The values of q([x_{i}]) and b([x_{i}], [x_{j}]) determine q by Definition 2.32. The constraint placed on the choice of q for odd order elements arises because q(nx) = n^{2}x (as we show below). If [x_{i}] is of odd order then p^{k}[x_{i}] = 0 for p an odd prime and some positive integer k. Thus 0 = p^{2k}q([x_{i}]) and hence the denominator of q([x_{i}]) must be odd. Now we show q(nx) = n^{2}q(x) since firstly

b([x], [x]) = −b([x], −[x])
= −(q(0) − q([x]) − q(−[x]))
= 2q([x]).

From this it follows that q((n + 1)[x]) − q(n[x]) − q([x]) = b(n[x], [x]) = 2nq([x]) and so by induction q(n[x]) = n^{2}q([x]).

Lemma 2.31. Let b : T G × T G → ℚ/ℤ be a linking form, let Q(b) be the set of all quadratic linking functions refining b and let q ∈ Q(b). For any a ∈ T G define

q_{a}(x) := q(x) + b(x, a) = q(x + a) − q(a).

Then, 1. q_{a} is a quadratic linking function refining b,

2. q_{a}(x) − q_{b}(x) = b(x, a − b),

3. T G acts freely and transitively on Q(b) by the action a · q = q_{a},

4. if q is homogeneous then q_{a} is also homogeneous if and only if 2a = 0, 5. if q = q_{a}^{o} where q^{o} is a quadratic linking form, then β(q) := 2a is independent

of the choice of q_{o} and a and thus an invariant of q.

Proof. (1.) Let x, y be elements of T G.

q_{a}(x + y) − q_{a}(x) − q_{a}(y) = q(x + y) + b(x + y, a)
−q(x) − b(x, a) − q(y) − b(y, a)
= q(x) + q(y) + b(x, y) − q(x) − q(y)
= b(x, y)

Thus q_{a} is a refinement of b. (2.) The second statement is clear. (3.) It is evident for a, b ∈ G that q_{a+b} = (q_{a})_{b} and hence we have an action. Suppose now that q′ is another quadratic refinement of b. We must show that q′ = q_{a} for some a ∈ T G hence that the action is transitive. Consider then the function on T G defined by r(x) = q′(x) − q(x). Clearly r(0) = 0 and

r(x + y) = q′(x + y) − q(x + y)
= q′(x) + q′(y) + b(x, y) − q(x) − q(y) − b(x, y)
= (q′(x) − q(x)) + (q′(y) − q(y))
= r(x) + r(y).

Thus r is linear and since b is nonsingular there exists an element a ∈ T G such that r(x) = b(x, a). So we see that

q_{a}(x) = q(x) + b(x, a)
= q(x) + r(x)
= q′(x).

Finally note that 2 and the nonsingularity of b entail that q_{a} = q_{b} if and only if a = b and hence the action is free.

(4.) Assume that q is homogeneous. Then

\begin{aligned}q_{a}(x) − q_{a}(−x) & = q(x) + b(x, a) − (q(−x) + b(−x, a)) \\ & = b(2x, a). \end{aligned}

Now b(2a, x) = 0 for all x ∈ G if and only if 2a = 0 and hence q_{a} is homogeneous if and only if 2a = 0. (5.) By Lemma 2.30, there is a quadratic form q^{o} which refines b and so by 3, there is an a such that q = q_{a}^{o}. Now let q^{o′} be another quadratic form refining b and let a′ be such that q = q_{a′}^{o′}. By 3 and 4, q^{o′} = q_{ϵ}^{o} where 2ϵ = 0 and by 3, a = a′ + ϵ. Thus 2a′ = 2a as required.

We now describe how a nondegenerate quadratic function κ(H, λ, α) induces a quadratic linking function on the torsion subgroup of G = Cok(λ̂). Recall the fundamental sequence of κ

0 ⟶ H ⟶^{λ̂} H^{∗} ⟶^{π} T G ⟶ 0

and that for x ∈ H^{∗} we denote π(x) by [x].

Definition 2.32. Let κ = κ(H, λ, α) be a nondegenerate quadratic function. If κ ∈ F^{ev} then λ is even and we define

\begin{aligned}q^{ev}(κ) : & T G & → & ℚ/ℤ \\ & [x] & ↦ & λ^{−1}(x, x)/2 mod ℤ. \end{aligned}

If κ ∈ F^{c} then we define

\begin{aligned}q^{c}(κ) : & T G & → & ℚ/ℤ \\ & [x] & ↦ & (λ^{−1}(x, x) + λ^{−1}(x, α))/2 mod ℤ. \end{aligned}

Remark 2.33. I am grateful to Stephan Stolz who first suggested the definition of q^{c} to me and also use of Gauss sums (see Chapter 5) in the classification of quadratic functions. The definition of q^{ev} has been know at least since [Wa3] and its properties

9. QUADRATIC LINKING FUNCTIONS 47

have been thoroughly investigated by Wall [Wa3], Durfee [Du1] and [Du2] and Nikulin [Ni]. Much of this thesis is devoted to proving that the results of these three authors concerning even quadratic forms and quadratic linking forms generalize to characteristic quadratic functions and quadratic linking functions.

Lemma 2.34. Let κ = κ(H, λ, α) be a nondegenerate quadratic function in F^{∗}.

1. If κ ∈ F^{ev} then q^{ev}(κ) is a homogeneous refinement of b(λ).

2. If κ ∈ F^{c} then q^{c}(κ) is a quadratic refinement of b(λ).

Proof. The calculations are entirely routine. Since definition the of q^{ev}(κ) is well known, we verify only that q^{c}(κ) is well defined. Thus let κ ∈ F^{∗}, let x ∈ H^{∗} and let x + λ̂(v) be another representative of [x]. We calculate that λ^{−1}(x + λ̂(v), x + λ̂(v) + α)/2 mod ℤ

= λ^{−1}(x, x + α)/2 + [2λ^{−1}(x, λ̂(v)) + λ^{−1}(λ̂(v), λ̂(v)) + λ^{−1}(λ̂(v), α)]/2 mod ℤ

= λ^{−1}(x, x + α)/2 + [2x(v) + λ(v, v) + α(v)]/2 mod ℤ
= λ^{−1}(x, x + α)/2 mod ℤ,

where the last equality holds because α is characteristic for λ.

Next, given x, y ∈ H^{∗} and [x], [y] ∈ T G we calculate that

q^{ev}(κ)([x] + [y]) = λ^{−1}(x + y, x + y)/2  mod ℤ
= λ^{−1}(x, x)/2 + λ^{−1}(y, y)/2 + λ^{−1}(x, y) mod ℤ
= q^{ev}(κ)([x]) + q^{ev}([y]) + b(λ)([x], [y]).

Hence q^{ev} refines b(λ). Moreover it is clear that q^{ev}(−[x]) = q^{ev}([x]) and hence we have proven 1. In the characteristic case we calculate that

q^{c}(κ)([x] + [y]) = λ^{−1}(x + y, x + y)/2 + λ^{−1}(x + y, α)/2 mod ℤ
= (λ^{−1}(x, x) + λ^{−1}(x, α))/2
+(λ^{−1}(y, y) + λ^{−1}(y, α))/2 + λ^{−1}(x.y) mod ℤ
= q^{c}(κ)([x]) + q^{c}([y]) + b(λ)([x], [y])

9. QUADRATIC LINKING FUNCTIONS 48

and this proves 2.

Given two quadratic linking functions q_{0} : T G_{0} → ℚ/ℤ and q_{1} : T G_{1} → ℚ/Z an isomorphism θ : T G_{0} → T G_{1} is called an isometry if q_{0} = q_{1} ∘ θ. In this situation we shall write that q_{0} ≅ q_{1}. We consider next the question of which quadratic linking functions are isometric to q^{∗}(κ) for some κ.

Theorem 2.35 (Wall [Wa3]). Let q : T G → ℚ/ℤ be a quadratic linking form on a finite abelian group. Then there is a nondegenerate even quadratic form κ(λ, 0) such that q ≅ δ^{ev}(κ).

Corollary 2.36. 1. Let b : T G × T G → ℚ/ℤ be a linking form on a finite

abelian group, then there is a bilinear form λ such that b ≅ b(λ).

2. Let q : T G → ℚ/ℤ be a quadratic linking function of a finite abelian group, then there is a characteristic quadratic function κ such that q ≅ δ^{c}(κ).

Proof. (1.) By Lemma 2.30 there is a quadratic linking form q which refines b. Now apply Theorem 2.35 to deduce that there is a nondegenerate quadratic form κ(λ, 0) such that q ≅ δ^{ev}(κ). From Lemma 2.34 we see that b ≅ b(λ). (2.) We begin by stating and proving a simple lemma which will be useful elsewhere.

Lemma 2.37. Let κ = κ(H, λ, α) be a nondegenerate characteristic quadratic function and let ϵ ∈ H^{∗} be divisible by 2. Then

q^{c}(κ(H, λ, α + ϵ)) = q^{c}(κ)_{[ϵ/2]}.

Proof. Let [x] be any element in T G = Cok(λ̂).

q^{c}(κ(λ, α + ϵ))([x]) = λ̂^{−1}(x, x + α + ϵ)/2  mod ℤ
= λ̂^{−1}(x, x + α)/2 + λ̂^{−1}(x, ϵ/2) mod ℤ
= q^{c}(κ)([x]) + b(λ)([x], [ϵ/2])
= q^{c}(κ)_{[ϵ/2]}([x]).

Now let b be the linking form induced by q and let q′ be a homogeneous refinement of b. By Theorem 2.35 there is an even quadratic form κ(H, λ, 0) and an isomorphism θ : Cok(λ̂) → T G such that q′ ∘ θ = δ^{ev}(κ). By Lemma 2.31 there is an element a ∈ T G such that q = q′_{a}. So choose ϵ ∈ H^{∗} such that θ([ϵ/2]) = a. As λ is even, κ(λ, 0) is characteristic and we may apply Lemma 2.37 to κ(λ, 0) and ϵ to conclude that δ^{c}(κ(λ, ϵ)) = q′_{a} ∘ θ = q ∘ θ.

Observe that any quadratic linking function q over a finite abelian group T G, has for invariants the bilinear for it refines, b(q), and the “linear element”, β(q) ∈ T G. We shall call the pair (b(q), β(q)) the “Wilkens invariants” of q. Chapter 5 deals with question of when quadratic functions q_{0} and q_{1} with isomorphic Wilkens invariants are isometric. (The answer is that there is one extra invariant, the Kervaire-Arf Gauss sum invariant K, such that q_{0} ≅ q_{1} if and only if (b(q_{0}), β(q_{0}) ≅ (b(q_{1}), β(q_{1})) and K(q_{0}) = K(q_{1})). We conclude this section with the following obvious but important remark concerning β(q).

Remark 2.38. Let q be a quadratic function on a finite abelian group T G. The element β(q) ∈ T G is even. If q = δ^{c}(κ), where κ is the nondegenerate quadratic function κ = κ(λ, α), then β(q) = [α] ∈ Cok(λ̂) = T G.

10. Quadratic linking families

In this section we define the notion of a quadratic linking family on a finitely generated abelian group G and describe how a quadratic function κ(λ, α) induces a quadratic linking family on its quotient group G = Cok(λ̂).

We begin with some terminology and preliminary definitions before our main definition. An element β in the abelian group G shall be called n-divisible for a natural number n if β = nγ for some γ ∈ G. If in addition γ is indivisible β shall

10. QUADRATIC LINKING FAMILIES 50

be called precisely n-divisible. We shall call 2-divisible elements even. Recall the homomorphism τ_{G} : G → G^{∗∗} from G to its double dual. Of course the torsion subgroup of G, T G, is the kernel of τ_{G}. The set of sections of τ_{G} is the set of homomorphisms

Sec(τ_{G}) := {Φ : G^{∗∗} → G|τ_{G} ∘ Φ = Id_{G^{∗∗}}}.

There is a free and transitive action of Hom(G^{∗∗}, T G) on Sec(τ_{G}) defined for ϕ ∈ Hom(G^{∗∗}, T G), Φ ∈ Sec(τ_{G}) and z ∈ G^{∗∗} as follows:

(ϕ · Φ)(z) = Φ(z) − ϕ(z).

Now for any linking form b defined on T G recall from Lemma 2.31 that T G acts freely and transitively on Q(b) which is the set of all quadratic linking functions refining b. Given an even element β ∈ G we let Hom(G^{∗∗}, T G) act on Q(b) by ϕ · q = q_{ϕ(τ_{G}(β)/2)}.

Definition 2.39. Let G be a finitely generated abelian group with torsion subgroup T G. A quadratic linking family on G is a triple (G, q^{∗}, β) where q^{∗} : Sec(τ_{G}) → Q(b) is a function for some linking form b defined on T G and β is an element of G.

1. A characteristic quadratic linking family on G is a quadratic linking family Q^{c} = (G, q^{c}, β) with β even and satisfying the following properties: (A) q^{c} is equivariant with respect to the actions of Hom(G^{∗∗}, T G), that is,

q^{c}(ϕ · Φ) = q^{c}(Φ)_{ϕ(τ_{G}(β)/2)} for every ϕ ∈ Hom(G^{∗∗}, T G) and every Φ ∈ Sec(τ_{G}),

(B) β = Φ(τ_{G}(β)) + β(q^{c}(Φ)) for every Φ ∈ Sec(τ_{G}), (recall that β(q^{c}(Φ)) is

the even element in T G defined in Lemma 2.31, 5).

2. An even quadratic linking family on G is a quadratic linking family Q^{ev} = (G, q^{ev}, β) where q^{ev} is a constant function whose value is a quadratic linking form refining b and β may be any element of G.

10. QUADRATIC LINKING FAMILIES 51

In order to motivate the somewhat complicated definition of a characteristic quadratic linking family we proceed now to show how a quadratic function induces a quadratic linking family on its quotient group. So suppose that κ(H, λ, α) is a possibly degenerate quadratic function with π : H^{∗} → G = Cok(λ̂) the projection onto the quotient of λ and with F = Ker(λ̂) the radical of λ. In Section 6 we defined the canonical projection ρ : H → H/F, the sections of ρ, Sec(ρ) = {Ψ : H/F → H|ρ ∘ Ψ = Id_{H/F} } and the projections p_{F} (Ψ) : H → F. Every section Ψ ∈ Sec(ρ) defines a section Φ(Ψ) ∈ Sec(τ_{G}) which is the composition

Φ(Ψ) := π ∘ p_{F} (Ψ)^{∗} ∘ λ̄_{F}^{∗} : G^{∗∗} ⟶^{λ̄_{F}^{∗}} F^{∗} ⟶^{p_{F} (Ψ)^{∗}} F^{∗}(Ψ) ⟶^{π|_{F^{∗}(Ψ)}} G

where λ̄_{F} was defined in Section 7. Now by Lemma 2.21 every x ∈ G^{∗} is of the form of λ̄_{F} (v) for some v ∈ F and for z ∈ G^{∗∗} we calculate that

τ_{G}(Φ(Ψ)(z))[x] = x[Φ(Ψ)(z)]
= λ̄_{F} (v)[(π|_{F^{∗}(Ψ)} ∘ p_{F} (Ψ)^{∗} ∘ λ̄_{F}^{∗})(z)]
= (p_{F} (Ψ)^{∗} ∘ λ̄_{F}^{∗} (z))[v]
= λ̄_{F}^{∗} (z)[p_{F} (Ψ)(v)]
= λ̄_{F}^{∗} (z)[v]
= z[λ̄_{F} (v)]
= z(x).

Hence τ_{G} ∘ Φ(Ψ) = Id_{G^{∗∗}} and Φ(Ψ) is indeed a section of τ_{G}. There is a free and transitive action of Hom(H/F, F) on Sec(ρ) and for ψ ∈ Sec(ρ) we define ϕ(ψ) ∈ Sec(τ_{G}) to be the composition

ϕ(ψ) : G^{∗∗} ⟶^{λ̄_{F}^{∗}} F^{∗} ⟶^{ψ^{∗}} (H/F)^{∗} ⟶^{π∘ρ^{∗}} T G.

Lemma 2.40. Consider the mappings

δ_{H} : Hom(H/F, F) → Hom(G^{∗∗}, T G)_{ψ  ↦  ϕ(ψ)} δ_{S} : Sec(ρ) → Sec(τ_{G})
Ψ ↦ Φ(Ψ).

10. QUADRATIC LINKING FAMILIES 52

1. δ_{H} is a surjective homomorphism,

2. for every Ψ ∈ Sec(ρ) and every ψ ∈ Hom(H/F, F), Φ(ψ · Ψ) = ϕ(ψ) · (Φ(Ψ)),

3. δ_{S} is onto and δ_{S}^{−1}(Φ(Ψ)) = {ψ · Ψ|ψ ∈ Ker(δ_{H})}.

Proof. (1.) We show first that δ_{H} is onto. Let {z_{1}, …, z_{l}} be a basis for G^{∗∗}. Every section ϕ ∈ Sec(τ_{G}) is determined by ϕ(z_{1}), …, ϕ(z_{l}). We choose y_{1}, …, y_{l} ∈ (H/F)^{∗} so that π ∘ ρ^{∗}(y_{i}) = ϕ(z_{i}) for each i which we may do since π ∘ ρ^{∗} is onto T G. Now as λ̄_{F}^{∗} : G^{∗∗} → F^{∗} is an isomorphism {λ̄_{F}^{∗} (z_{1}), …, λ̄_{F}^{∗} (z_{l})} is a basis for F^{∗} and we choose ψ : H/F → F so that ψ^{∗}(λ̄_{F}^{∗} (z_{i})) = y_{i} for each i. It follows that

ϕ(ψ)(z_{i}) = (π ∘ ρ^{∗})(ψ^{∗}(λ̄_{F}^{∗} (z_{i})) = ϕ(z_{i})

for each i. Thus ϕ(ψ) = ϕ and so δ is onto.

The group operation in both Hom(H/F, F) and Hom(G^{∗∗}, T G) is pointwise addition. For ψ_{0}, ψ_{1} ∈ Hom(H/F, F),

ϕ(ψ_{0} + ψ_{1}) = π ∘ ρ^{∗} ∘ (ψ_{0}^{∗} + ψ_{1}^{∗}) ∘ λ̄_{F}^{∗}
= π ∘ ρ^{∗} ∘ ψ_{0}^{∗} ∘ λ̄_{F}^{∗} + π ∘ ρ^{∗} ∘ ψ_{1}^{∗} ∘ λ̄_{F}^{∗}
= ϕ(ψ_{0}) + ϕ(ψ_{1}).

(2.) Observe first that for any y ∈ F^{∗} and any v ∈ H

p_{F} (ψ · Ψ)^{∗}(y)(v) = y(v − (Ψ + ψ)(ρ(v)))
= y(v − Ψ(ρ(v))) − y(ψ(ρ(v)))
= p_{F} (Ψ)^{∗}(y)(v) − ρ^{∗}(ψ^{∗}(y))(v).

We therefore calculate that for z ∈ G^{∗∗}

Φ(ψ.Ψ)(z) = (π|_{F (ψ.Ψ)})[p_{F} (ψ.Ψ)^{∗}(λ̄_{F}^{∗} (z))]
= π[p_{F} (Ψ)^{∗}(λ̄_{F}^{∗} (z)) − ρ^{∗}(ψ^{∗}(λ̄_{F}^{∗} (z)))]
= Φ(Ψ)(z) − ϕ(ψ)(z)
= (ϕ(ψ) · Φ(Ψ))(z).

(3.) This follows immediately from the 1, 2 and the fact that Hom(H/F, F) and Hom(G^{∗∗}, T G) act freely and transitively on Sec(ρ) and Sec(τ_{G}) respectively.

Now recall from proposition Proposition 2.18 that every section Ψ ∈ Sec(ρ) defines a splitting κ = α|_{F} ⊕ κ(Ψ) where κ(Ψ) = κ|_{Ψ(H)} is nondegenerate. Recall also Definition 2.32 which defined, for a nondegenerate quadratic function κ′(λ′, α′) ∈ F^{∗}, a quadratic linking function q^{∗}(κ′) on Cok(λ̂′).

Lemma 2.41. Let κ = κ(H, λ, α) be a quadratic function and let β = [α] ∈ Cok(λ̂). Then, for every ψ ∈ Hom(H/F, F) and every Ψ ∈ Sec(ρ),

1. if κ ∈ F^{ev}, q^{ev}(κ(ψ · Ψ)) = q^{ev}(κ(Ψ)),

2. if κ ∈ F^{c}, q^{c}(κ(ψ · Ψ)) = q^{c}(κ(Ψ))_{a}, where a = ϕ(ψ)(τ_{G}(β))/2 ∈ T G.

Proof. Each q^{∗}(κ(Ψ)) is defined on T G = Cok(λ̂(Ψ)) = H^{∗}(Ψ)/λ̂(Ψ)(H(Ψ)) and by Proposition 2.18 both H^{∗}(Ψ) and Cok(λ̂(Ψ)) = T G are independent of Ψ. By Lemma 2.20 the bilinear pairing λ(Ψ)^{−1} : H^{∗}(Ψ) × H^{∗}(Ψ) → ℚ is also independent of Ψ. Now for x ∈ H^{∗}(Ψ) and [x] = π(x) ∈ T G,

q^{ev}(κ(Ψ))([x]) = λ(Ψ)^{−1}(x, x)/2 mod ℤ

and so q^{ev}(κ(Ψ)) is independent of Ψ and 1 is proven. (2.) By the preceding remarks the only effect that altering Ψ may have on q^{c}(κ(Ψ)) comes via altering the linear part of κ(Ψ) which is α(Ψ). Now by 5 of Proposition 2.18 α(ψ · Ψ) = α(Ψ) + ρ^{∗}(ψ^{∗}(α|_{F})). Since κ is characteristic α|_{F} must be even for if v ∈ F, 0 = λ(v, v) = α(v) (mod 2). We may therefore apply Lemma 2.37 to conclude that

q^{c}(κ(ψ · Ψ)) = q^{c}(κ(λ(Ψ), α(Ψ) + ρ^{∗}(ψ^{∗}(α|_{F}))) = q^{c}(κ(Ψ))_{a}

10. QUADRATIC LINKING FAMILIES 54

where a = π(ρ^{∗}(ψ^{∗}(α|_{F})))/2 and π, as usual, is the projection H^{∗} → Cok(λ̂). Now since π(α) = β, α|_{F} = λ̄_{G}(β) and by Lemma 2.21, λ̄_{G} = λ̄_{F}^{∗} ∘ τ_{G}. Therefore

a = π(ρ^{∗}(ψ^{∗}(α|_{F})))/2
= (π ∘ ρ^{∗} ∘ ψ^{∗} ∘ λ̄_{F}^{∗})(τ_{G}(β))/2
= ϕ(ψ)(τ_{G}(β))/2.

Proposition 2.42. Let κ = κ(H, λ, α) be a quadratic function in F^{∗} with G = Cok(λ̂) and F = Ker(λ̂). Let τ_{G} : G → G^{∗∗}, Sec(τ_{G}), ρ : H → H/F and Sec(ρ) be as above and let β := [α] ∈ G. If κ ∈ F^{∗} and q^{∗}(κ) is defined by

\begin{aligned}q^{∗}(κ) : & Sec(τ_{G}) & → & Q(b(λ)) \\ & Φ & ↦ & q^{c}(κ(Ψ)) \end{aligned}

for any Ψ ∈ Sec(ρ) such that Φ(Ψ) = Φ. Then the triple

δ^{∗}(κ) := (G, q^{∗}(κ), β)

is a characteristic (resp. even) quadratic linking family on G for ∗ = c (resp. ∗ = ev).

Proof. Suppose that κ ∈ F^{c}. We show first the q^{c}(κ) is well defined. By Lemma 2.34 each q^{c}(κ(Ψ)) refines b(λ(Ψ)) which is independent of Ψ and equal, by definition, to b(λ). Secondly, by Lemma 2.40, if Ψ and Ψ′ are such that Φ(Ψ) = Φ(Ψ′) then Ψ′ = ψ · Ψ′ for some ψ such that ϕ(ψ) = 0. If follows now from 2 of Lemma 2.41 that q^{c}(κ(Ψ′)) = q^{c}(κ(Ψ)) and so q^{c}(κ) is well defined. Now let Φ ∈ Sec(τ_{G}) and ϕ ∈ Hom(G^{∗∗}, T G) and let Ψ ∈ Sec(ρ) and ψ ∈ Hom(H/F, F) be such that Φ(Ψ) = Φ and ϕ(ψ) = ϕ.

q^{c}(ϕ · Φ) = q^{c}(κ(ψ.Ψ))  by Lemma 2.40 2
= q^{c}(Ψ)_{ϕ(τ_{G}(β))/2} by Lemma 2.41 1.

Hence q^{c}(κ) is equivariant with respect to the action of Hom(G^{∗∗}, T G). With Φ and Ψ as before we must show that β = Φ(τ_{G}(β))+β(q^{c}(Φ)). Recall from the proof of Lemma

10. QUADRATIC LINKING FAMILIES 55

2.41 that α|_{F} = λ̄_{F}^{∗} (τ_{G}(β)) and also from Section 6 that α = p_{F} (Ψ)^{∗}(α|_{F}) + α(Ψ). We calculate that

β = π(α)
= π(p_{F} (Ψ)^{∗}(α|_{F}))) + π(α(Ψ))
= (π ∘ p_{F} (Ψ)^{∗} ∘ λ̂_{F}^{∗})(τ_{G}(β)) + β(q^{c}(κ(Ψ)))
= Φ(τ_{G}(β)) + β(q^{c}(Φ)).

Finally, this decomposition shows that β is even for by Remark 2.38, β(q^{c}(Ψ)) is even and, as we showed in the proof of Lemma 2.41, α|_{F} (and hence τ_{G}(β)) is even.

If κ ∈ F^{ev} then by Lemma 2.41, q^{ev}(κ(Ψ)) is independent of Ψ and hence q^{ev}(κ) is a constant function. By Lemma 2.34 q^{ev}(κ) is a homogeneous refinement of b(λ) and so the triple (G, q^{ev}(κ), β) defines an even quadratic linking family on G.

Remark 2.43. We note that we have been using the symbol q^{∗} in two different but closely related ways. When κ = κ(λ) is a degenerate quadratic function the symbol q^{∗}(κ) denotes a function which is defined over Sec(τ_{G}) (or equivalently the splittings of G = Cok(λ̂)) which values in the quadratic linking functions defined on T G. When κ = κ(λ) is a nondegenerate characteristic quadratic function the symbol q^{∗}(κ) denotes a single quadratic linking function define on T G = Cok(λ̂). Now κ is nondegenerate if and only if G = T G is torsion. In this case G^{∗∗} = {e} and Sec(τ_{G}) = {0} is a singleton set containing the zero homomorphism from the trivial group to T G. Hence our two uses of q^{∗} coincide provided we are prepared to identify a function on a singleton set with its range. In the characteristic case observe also that when G = T G is a torsion group β = β(q^{c}(0)) is determined by q^{c}(0) and so in the torsion case a characteristic quadratic linking family is completely determined by q^{c}(0).

Remark 2.44. Note that the equivariance property (B) of characteristic quadratic linking families ensures that the function q^{c} is determined by its value on any

10. QUADRATIC LINKING FAMILIES 56

single section Φ ∈ Sec(τ_{G}) and the divisibility of τ_{G}(β). In particular if τ_{G}(β) is precisely 2r-divisible then a simple argument concerning the action of Hom(G^{∗∗}, T G) on Q(b) shows for any Φ ∈ Sec(τ_{G}) that Im(q^{c}) = {q^{c}(Φ)_{ra}|a ∈ T G}.

Remark 2.45. When G is a free group then T G = {e}, Sec(τ_{G}) = {Id_{G}} and q^{c}(Id) is the trivial quadratic linking function.

We now define isometries between and sums of quadratic linking functions. Let Q_{0} = (G_{0}, q_{0}^{∗}, β_{0}) and Q_{1} = (G_{1}, q_{1}^{∗}, β_{1}) be a pair of quadratic linking families. An isomorphism θ : G_{0} → G_{1} defines a bijection

τ_{θ} : Sec(τ_{G_{0}}) →  Sec(τ_{G_{1}})
Φ  ↦ θ ∘ Φ ∘ (θ^{∗∗})^{−1}

which is illustrated in the following commutative diagram.

[figure: G_{0}; G_{1}; G_{0}^{∗∗}; G_{1}^{∗∗}; θ; Φ; τ_{θ}(Φ); (θ^{∗∗})^{−1}]

We define θ to be an isometry from Q_{0} to Q_{1} if θ(β_{0}) = β_{1} and if for all Φ ∈ Sec(τ_{G_{0}}), q_{0}^{∗}(Φ) = q_{1}^{∗}(τ_{θ}(Φ)) ∘ θ_{T G_{0}}.

Remark 2.46. If If Q_{0} and Q_{1} are characteristic quadratic linking families then Q_{0} and Q_{1} are isometric if and only if there is an isomorphism θ : G_{0} → G_{1} such that θ(β_{0}) = β_{1} and q_{1}^{c}(Φ_{1}) ∘ θ_{T G_{0}} = q_{0}^{c}(Φ_{0}) for some sections Φ_{0} ∈ Sec(τ_{G_{0}}) and Φ_{1} ∈ Sec(τ_{G_{1}}). However, there is no guarantee that θ itself is an isometry from Q_{0} to Q_{1}. But there is an automorphism θ′ : G_{1} ≅ G_{1} such that θ′|_{T G_{1}} = Id_{T G_{1}} and such that θ′ ∘ θ is an isometry from Q_{0} to Q_{1}.

Given homomorphisms f_{0} : G_{0} → F_{0} and f_{1} : G_{1} → G_{0} we let f_{0} ⊕ f_{1} denote the obvious homomorphism from G_{0} ⊕ G_{1} to F_{0} ⊕ F_{1}. It is clear that τ_{G_{0}⊕G_{1}} = τ_{G_{0}} ⊕ τ_{G_{1}}

10. QUADRATIC LINKING FAMILIES 57

and that there is an inclusion

Sec(τ_{G_{0}}) ⊕ Sec(τ_{G_{1}}) : → Sec(τ_{G_{0}⊕G_{1}})
(Φ_{0}, Φ_{1})  ↦ Φ_{0} ⊕ Φ_{1}.

With the quadratic linking families Q_{0} and Q_{1} as above we define Q_{0} ⊕ Q_{1} to be the quadratic linking family (G_{0} ⊕ G_{1}, q_{0}^{∗} ⊕ q_{1}^{∗}, (β_{0}, β_{1})). Here

q_{0}^{∗} ⊕ q_{1}^{∗} : Sec(τ_{G_{0}⊕G_{1}}) → Q(b_{0} ⊕ b_{1})

is the unique function such that (q_{0}^{∗} ⊕ q_{1}^{∗})(Φ_{0} ⊕ Φ_{1}) = q_{0}^{∗}(Φ_{0}) ⊕ q_{1}^{∗}(Φ_{1}) for every Φ_{i} ∈ Sec(τ_{G_{i}}), i = 0, 1 and which is equivariant with respect to the action of Hom(G_{0}^{∗∗} ⊕ G_{1}^{∗∗}, T G_{0} ⊕ T G_{1}). Note that in the even case we take this action to be trivial so that q_{0}^{ev} ⊕ q_{1}^{ev} is again a constant function.

We now show that the isometries and directs sums of quadratic functions induce isometries and direct sums of the associated quadratic linking families. Let κ_{0} = (H_{0}, λ_{0}, α_{0}) and κ_{1} = (H_{1}, λ_{1}, α_{1}) be a pair of quadratic functions with canonical projections ρ_{i} : H_{0} → H_{0}/F_{0}, i = 0, 1. For every isometry Θ : H_{0} → H_{1}, Θ(F_{0}) = F_{1} and hence Θ induces an isomorphism Θ_{/F} : H_{0}/F_{0} → H_{1}/F_{1}. The isometry Θ defines a bijection

ρ_{Θ} : Sec(ρ_{0}) →  Sec(ρ_{1})
Ψ ↦ Θ ∘ Ψ ∘ (Θ_{/F})^{−1}

which is illustrated in the following commutative diagram.

[figure: H_{0}; T_{1}; H_{0}/F_{0}; H_{1}/F_{1}; Θ; Ψ; ρ_{Θ}(Ψ); (Θ_{/F})^{−1}]

Lemma 2.47. Let Θ : H_{0} → H_{1} be an isometry between quadratic functions κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}). Then, 1. the following diagram commutes,

[figure: 0; F_{0}; H_{0}; H_{0}^{∗}; G_{0}; 0; 0; F_{1}; H_{1}; H_{1}^{∗}; G_{1}; 0; Θ|_{F_{0}}; λ̂_{0}; Θ; π_{0}; (Θ^{∗})^{−1}; λ̂_{1}; π_{1}]

2. for x ∈ H_{0}^{∗} and [x] = π_{0}(x) ∈ G_{0}, the map

\begin{aligned}δ^{∗}(Θ) : & G_{0} & → & G_{1} \\ & [x] & ↦ & [(Θ^{∗})^{−1}x] \end{aligned}

is a well-defined isomorphism from G_{0} to G_{1},

3. for every Ψ ∈ Sec(ρ_{0}), Φ(ρ_{Θ}(Ψ)) = τ_{δ^{∗}(Θ)}(Φ(Ψ)) ∈ Sec(τ_{G_{1}}),

4. for every Ψ ∈ Sec(ρ_{0}), Θ|_{H(Ψ)} : κ_{0}(Ψ) → κ_{1}(ρ_{Θ}(Ψ)) is an isometry,

5. if κ_{0} and κ_{1} are nondegenerate then λ̂_{0}^{−1} = (Θ^{−1} ⊗ Id_{ℚ}) ∘ λ̂_{1}^{−1} ∘ (Θ^{∗})^{−1}.

Proof. (1.) Let v_{0}, w_{0} be elements of H_{0}.

Θ^{∗}(λ̂_{1}(Θ(v_{0})))(w_{0}) = λ̂_{1}(Θ(v_{0}))(Θ(v_{1}))
= λ_{1}(Θ(v_{0}), Θ(v_{1}))
= λ_{0}(v_{0}, w_{0})
= λ̂_{0}(v_{0})(w_{0})

Thus λ̂_{0} = Θ^{∗} ∘ λ̂_{1} ∘ Θ and so (Θ^{∗})^{−1} ∘ λ̂_{0} = λ̂_{1} ∘ Θ and the diagram commutes. (2.) This follows from part 1 and an elementary diagram chase. (3.) This is a particularly complicated but routine diagram chase. We include the commutative diagram and leave the reader to check that it commutes.

[figure: G_{0}^{∗∗}; F_{0}^{∗}; F_{0}^{∗}(Ψ); G_{0}; G_{1}^{∗∗}; F_{1}^{∗}; F_{1}^{∗}(ρ_{Θ}(Ψ)); G_{1}; λ̄_{F_{0}}^{∗}; p_{F_{0}}(Ψ)^{∗}; π_{0}|_{F^{∗}_{0}(Ψ)}; (Θ^{∗})^{−1}|_{F^{∗}_{0}(Ψ)}; δ^{∗}(Θ); λ̄_{F_{1}}^{∗}; δ^{∗}(Θ)^{∗∗}; p_{F_{1}}(ρ_{Θ}(Ψ))^{∗}; (Θ|_{F_{0}})^{∗}; π_{1}|_{F^{∗}_{1}(ρ_{Θ}(Ψ))}]

The homomorphism Φ(ρ_{Θ}(Ψ)) is the composition of the bottom three horizontal arrows whereas the homomorphism τ_{δ^{∗}(Θ)}(Φ(Ψ)) moves up δ^{∗}(Θ)^{∗∗} along the top of the diagram and then down δ^{∗}(Θ).

(4.) It is only necessary to observe that

Θ(H_{0}(Ψ)) = Θ(Ψ(H_{0}/F_{0})) = ρ_{Θ}(Ψ)(Θ_{/F} (H_{0}/F_{0})) = ρ_{Θ}(Ψ)(H_{1}/F_{1}) = H_{1}(ρ_{Θ}(Ψ)),

because the restriction of an isometry remains an isometry. (5.) If κ_{0} and κ_{1} are nondegenerate then λ̂_{i}⊗Id_{ℚ} is invertible and the homomorphism λ̂_{i}^{−1} = (λ̂_{i} ⊗ Id_{ℚ})^{−1}|_{H_{i}^{∗}}, i = 0, 1. We calculate that

λ̂_{0}^{−1} = ((Θ^{∗} ∘ λ̂_{1} ∘ Θ) ⊗ Id_{ℚ})^{−1}|_{H_{0}^{∗}}
= (Θ^{−1} ⊗ Id_{ℚ}) ∘ (λ̂_{1}^{−1} ⊗ Id_{ℚ})|_{H_{1}^{∗}} ∘ ((Θ^{∗})^{−1} ⊗ Id_{ℚ})|_{H_{0}^{∗}}
= (Θ^{−1} ⊗ Id_{ℚ}) ∘ λ̂_{1}^{−1} ∘ (Θ^{∗})^{−1}.

Lemma 2.48. Let κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}) be a pair of quadratic functions in F^{∗}.

1. If Θ : κ_{0} → κ_{1} is an isometry then δ^{∗}(Θ) : δ^{∗}(κ_{0}) → δ^{∗}(κ_{1}) is an isometry of quadratic linking families.

2. δ^{∗}(κ_{0} ⊕ κ_{1}) = δ^{∗}(κ_{0}) ⊕ δ^{∗}(κ_{1}).

Proof. (1.) Let δ^{∗}(κ_{0}) = (G_{0}, q_{0}^{∗}, β_{0}) to δ^{∗}(κ_{1}) = (G_{1}, q_{1}^{∗}, β_{1}). As (Θ^{∗})^{−1}(α_{0}) = α_{1} it follows that

δ^{∗}(Θ)(β_{0}) = δ^{∗}(Θ)([α_{0}]) = [α_{1}] = β_{1}.

Suppose now that κ_{0} and κ_{1} are nondegenerate. Another diagram chase shows that δ^{∗}(Θ) is an isometry from δ^{∗}(κ_{0}) to δ^{∗}(κ_{1}). We include the computation for the characteristic case with x ∈ H^{∗} a lift of [x] ∈ T G_{0}.

q_{1}(δ^{c}(Θ)([x])) = λ_{1}^{−1}((Θ^{∗})^{−1}(x), (Θ^{∗})^{−1}(x) + α_{1})/2  mod ℤ
= (Θ^{∗})^{−1}(x + α_{0})[λ̂_{1}^{−1}((Θ^{∗})^{−1}(x))]/2  mod ℤ
= (Θ^{∗})^{−1}(x + α_{0})[(Θ ⊗ Id_{ℚ})(λ̂_{0}^{−1}(x))]/2 mod ℤ by Lemma 2.48, 5
= (x + α_{0})[λ̂_{0}^{−1}(x)]/2  mod ℤ
= λ_{0}^{−1}(x, x + α_{0})/2  mod ℤ
= q_{0}([x])

Now consider the general case. Let Ψ ∈ Sec(ρ_{0}) and Φ ∈ Sec(τ_{G_{0}}) be such that Φ = Φ(Ψ). By part 4 of Lemma 2.47, Θ|_{H(Ψ)} defines an isometry form κ_{0}(Ψ) to κ_{1}(ρ_{Θ}(Ψ)). Moreover, δ^{∗}(Θ)|_{T G_{0}} = δ^{∗}(Θ|_{H(Ψ)}) and so by the calculation for the nondegenerate case we conclude that q^{∗}(κ_{1}(ρ_{Θ}(Ψ))) ∘ δ^{∗}(Θ)|_{T G_{0}} = q^{∗}(κ_{0}(Ψ)). We calculate that

q_{1}^{∗}(τ_{δ^{∗}(Θ)}(Φ)) ∘ δ^{∗}(Θ)|_{T G_{0}} = q_{1}^{∗}(Φ(ρ_{Θ}(Ψ))) ∘ δ^{∗}(Θ)|_{T G_{0}} by Lemma 2.47 3
= q^{∗}(κ_{1}(ρ_{Θ}(Ψ))) ∘ δ^{∗}(Θ)|_{T G_{0}}
= q^{∗}(κ_{0}(Ψ))
= q_{0}^{∗}(Φ)

and so δ^{∗}(Θ) defines an isometry from δ^{∗}(κ_{0}) to δ^{∗}(κ_{1}). (2.) It is clear that Cok(λ̂_{0}⊕λ̂_{1}) = Cok(λ̂_{0})⊕Cok(λ̂_{1}) and that [(α_{0}, α_{1})] = ([α_{0}], [α_{1}]). It is also clear that the quadratic linking function induced by the direct sum of nondegenerate quadratic functions is the direct sum of the induced quadratic linking functions. Now we note that for Ψ_{0} ∈ Sec(ρ_{0}) and Ψ_{1} ∈ Sec(ρ_{1}) that Φ(Ψ_{0} ⊕ Ψ_{1}) =

10. QUADRATIC LINKING FAMILIES 61

Φ(Ψ_{0}) ⊕ Φ(Ψ_{1}). Setting Φ_{0} = Φ(Ψ_{0}) and Φ_{1} = Φ(Ψ_{1}) we calculate that

q^{∗}(κ_{0} ⊕ κ_{1})(Φ_{0} ⊕ Φ_{0}) = q^{∗}(κ_{0} ⊕ κ_{1})(Ψ_{0} ⊕ Ψ_{1})
= q^{∗}((κ_{0} ⊕ κ_{1})(Ψ_{0} ⊕ Ψ_{1}))
= q^{∗}(κ_{0}(Ψ_{0}) ⊕ κ_{1}(Ψ_{1}))
= q^{∗}(κ_{0}(Ψ_{0})) ⊕ q^{∗}(κ_{1}(Ψ_{1}))
= q^{∗}(κ_{0})(Φ_{0}) ⊕ q^{∗}(κ_{0})(Φ_{1})

which shows that δ^{∗}(κ_{0} ⊕ κ_{1}) = δ^{∗}(κ_{0}) ⊕ δ^{∗}(κ_{1}).

Remark 2.49. For i = 0, 1, let L_{i} = L(κ_{i}) be handlebodies corresponding to quadratic functions κ_{i} and let P_{i} = ∂L_{i}. The topological correlates of H_{i}, H_{i}^{∗} and G_{i} = Cok(λ̂_{i}) are respectively H_{4j}(L_{i}), H^{4j}(L_{i}) and H^{4j}(P_{i}). The isometry Θ : H_{0} → H_{1} therefore defines an isomorphism H_{4j}(L_{0}) → H_{4j}(L_{1}) which, by Wall’s classification of handlebodies, is realized as the map induced on homology by a diffeomorphism g : L_{0} → L_{1}. If f := g|_{P_{0}} is restriction of g to the boundary of L_{0}, then δ^{∗}(Θ) corresponds to the the “wrong way” map f_{!} : H^{4j}(P_{0}) → H^{4j}(P_{1}). Recall that f_{!} was defined in the introduction as follows. For x ∈ H^{4j}(P_{0}), f_{!}(x) := f_{∗}(x ∩ [P_{0}])/(P_{1}), where [P_{0}] ∈ H_{8j}(P_{0}) and (P_{1}) ∈ H^{8j}(P_{1}) are the orientation classes of P_{0} and P_{1} and ∩ and / denote the cap and slant products respectively.

Definition 2.50. Let P be a highly connected (8j − 1)-manifold and let L be a handlebody with boundary P #Σ_{P}. We define the quadratic linking family of P, Q(P), to be the quadratic linking family induced by κ(L),

Q(P) := δ^{∗}(κ(L)).

Lemma 2.51. The quadratic linking family of a highly connected manifold P is well defined and if P_{0} and P_{1} are two highly connected (8j − 1)-manifolds then

Q(P_{0}#P_{1}) = Q(P_{0}) ⊕ Q(P_{1}).

10. QUADRATIC LINKING FAMILIES 62

Proof. Note that the proof is completely analogous to the proof of Corollary 2.26. Suppose that L and L′ are two handlebodies with boundary P #Σ_{P}. Then by Wilkens’ Theorem 2.24 applied to Id_{P #Σ_{P}} there are handlebodies V and V′ with boundaries the standard sphere and a diffeomorphism

g : L♮V → L′♮V′

which extends Id_{P #Σ_{P}}. The induced map g_{∗} : κ(L) ⊕ κ(V) → κ(L′) ⊕ κ(V′) is an isometry. So by Lemma 2.48

δ^{∗}(g_{∗}) : δ^{∗}(κ(L)) ⊕ δ^{∗}(κ(V)) = δ^{∗}(κ(L′)) ⊕ δ^{∗}(κ(V′))

is an isometry. Now by the preceding remark, δ^{∗}(g_{∗}) = (g|_{∂(L♮V)})_{!} = (Id_{P #Σ_{P}})_{!} = Id. Moreover, both δ^{∗}(κ(V)) and δ^{∗}(κ(V′)) are the trivial quadratic linking family and hence δ^{∗}(κ(L)) = δ^{∗}(κ(L′)) and so Q(P) is well defined.

Now suppose that P_{i}#Σ_{i} = ∂L_{i} for i = 0, 1 then (P_{0}#P_{1})#Σ_{P_{0}#P_{1}} = ∂(L_{0}♮L_{1}) and we see that

Q(P_{0}#P_{1}) = δ^{∗}(κ(L_{0}♮L_{1}))
= δ^{∗}(κ(L_{0}) ⊕ κ(L_{1}))
= δ^{∗}(κ(L_{0})) ⊕ δ^{∗}(κ(L_{1}))
= Q(P_{0}) ⊕ Q(P_{1}).

Remark 2.52. In dimensions (8j−1) with j > 2, Wall gave an intrinsic definition of the quadratic form associated to a highly connected manifold P [Wa3]. At this stage I am still unable to give an intrinsic definition of Q(P) in dimensions 7 and 15 although this would be most desirable.

To conclude this section we summarize the algebraic invariants arising from a pair (L, P) which we have described in previous sections.

11. CATEGORICAL STATEMENTS 63

• A free abelian group H = H_{4j}(L) ≅ H^{4j}(L, P) and its dual H^{∗} = H^{4j}(L) ≅ H_{4j}(L, P).

• A symmetric intersection pairing λ : H×H → ℤ with adjoint map λ̂ : H → H^{∗}, λ(v, w) = λ(w, v) = λ̂(w)(v) = λ̂(v)(w).

• An element α ∈ H^{∗} which is the stable tangential invariant of L. When j = 1, 2 α is characteristic for λ, that is λ(v, v) + α(v) is even for all v ∈ H. When j > 2 there is no constraint upon α.

• A quadratic function κ = κ(H, λ, α) = κ(L) which is a synthesis of the above invariants.

• A finitely generated abelian group G = H^{4j}(P) ≅ H_{4j−1}(P).

• A nonsingular, symmetric, bilinear (linking) form b on T G, b : T G × T G → ℚ/ℤ.

• An element β ∈ G which is the stable tangential invariant of P. When j = 2, 4 β is even.

• A function q^{∗} : Sec(τ_{G}) → Q(b) which associates a quadratic linking function to every section of the canonical map τ_{G} : G → G^{∗∗} (in the even case q^{ev} is a constant function with value an quadratic linking form).

• A quadratic linking family, Q(P) = (G, q^{∗}, β) = δ^{∗}(κ(L)).

11. Categorical statements

Let HC^{8j−1} be the category whose objects are closed smooth highly connected (8j − 1)-manifolds. An almost diffeomorphism is a homeomorphism which fails to be a diffeomorphism at at most finitely many points (see Section 1). The morphisms of HC^{8j−1} shall be equivalence classes of almost diffeomorphisms f : P_{0} → P_{1}. Two almost diffeomorphisms f and f′ are equivalent if they induce the same isomorphism on cohomology, f_{!} = f′_{!} : H^{4j}(P_{0}) → H^{4j}(P_{1}). We let Q^{c} (resp. Q^{ev}) denote the

11. CATEGORICAL STATEMENTS 64

categories whose objects are characteristic (resp. even) quadratic linking families. The morphisms of Q^{∗} are isometries of quadratic linking families. The categories HC^{8j−1} and Q^{∗} are symmetric monoidal categories with respect to the operations of connected sum of manifolds and direct sum of quadratic linking families respectively.

Theorem 2.53. The functors Q^{j} which associate to every highly connected (8j − 1)-manifold its quadratic linking family and to every almost diffeomorphism the induced isomorphism on cohomology are equivalences of (symmetric monoidal) categories.

\begin{matrix}Q^{j} & : HC^{8j−1} & → & Q^{c} \\ & \{P, [f]\} & ↦ & \{Q(P), f_{!}\} \end{matrix} (j = 1, 2) \begin{matrix}Q^{j} & : HC^{8j−1} & → & Q^{ev} \\ & \{P, [f]\} & ↦ & \{Q(P), f_{!}\} \end{matrix} (j > 2)

Proof. Let \underline{HC}^{8j−1} be a skeleton of HC^{8j−1}. By Theorem B, Q^{j}(\underline{HC}^{8j−1}) is a skeleton of Q^{∗} and hence Q^{j} defines an equivalence of categories. Moreover, for any pair of highly connected (8j − 1)-manifolds, P_{0} and P_{1} and any pair of almost diffeomorphisms f_{0} and f_{1}, Q(P_{0}#P_{1}) = Q(P_{0}) ⊕ Q(P_{1}) and (f_{0}#f_{1})_{!} = (f_{0})_{!} ⊕ (f_{1})_{!}.

Of course the equivalences of categories in the restatement of Wall’s classification of handlebodies (2.12) are related to the equivalences of categories in Theorem 2.53. Recall that the functor κ^{j} from Theorem 2.12 associates to every 8j-dimensional handlebody its quadratic function and to every equivalence class of diffeomorphisms the induced map on homology. We let ∂^{j} : H^{8j} → HC^{8j−1} be the functor which associates to every handlebody its boundary and to the every equivalence class of diffeomorphisms the equivalence class of diffeomorphisms restricted to the boundary.

12. THE INVARIANTS s_{1}, s̄_{1} AND μ 65

Proposition 2.54. The functors ∂^{j}, δ^{∗}, κ^{j} and Q^{j} fit into the following commuting squares.

\begin{matrix}H^{8j} & ⟶^{κ^{j}} & F^{c} \\ ↓ ∂^{j} & & ↓ δ^{c} \\ HC^{8j−1} & ⟶^{Q^{j}} & Q^{c} \end{matrix} (j = 1, 2) \begin{matrix}H^{8j} & ⟶^{κ^{j}} & F^{ev} \\ ↓ ∂^{j} & & ↓ δ^{ev} \\ HC^{8j−1} & ⟶^{Q^{j}} & Q^{ev} \end{matrix} (j > 2).

Proof. This follows from the definition of Q(∂L) since Q(∂L) = δ^{∗}(κ(L)).

12. The invariants s_{1}, s̄_{1} and μ

Up until this point our preliminary remarks have been preparing for the almost diffeomorphism classification of highly connected (8j−1)-manifolds by their quadratic linking families. In this section we define the smooth invariant s_{1} of highly connected rational homology spheres in preparation for their smooth classification. We also define the almost diffeomorphism invariant s̄_{1} whose algebraic analogue is crucial for the classification of quadratic linking functions.

If a handlebody L has a nondegenerate quadratic function then the boundary of L, P, is a rational homology sphere bounding the a spin manifold in which case the Eels-Kuiper μ-invariant [EK] is well defined. In fact, Stolz [St] has defined a stronger invariant but only for homotopy spheres. However, there is nothing which prevents the definition of this invariant, which I shall call s_{1}, for simply connected rational homology spheres and so we now present a generalization of Stolz’s s_{1} invariant. Suppose then that M is a simply connected spin rational homology sphere of dimension 4k − 1 bounding a spin manifold W. We let L(W), Â(W) and ph(W) denote respectively the L-genus, the Â-genus and the Pontryagin character of W which are rational power series in the Pontryagin classes p_{i} of W, [Hi] and [St]. For any power series T in the Pontryagin classes we let T_{k} denote its 4k-dimensional component. So for

12. THE INVARIANTS s_{1}, s̄_{1} AND μ 66

example, the first four terms of the power series of L(W), Â(W) and ph(W) are

L_{0} = 1, L_{1} = \frac{1}{3}p_{1}, L_{2} = \frac{1}{45}(7p_{2} − p_{1}^{2}), L_{3} = \frac{1}{3^{2}·5·7}(62p_{3} − 13p_{2}p_{1} + 2p_{1}^{3}),
Â_{0} = 1, Â_{1} = −\frac{2}{3}p_{1}, Â_{2} = \frac{2}{45}(−4p_{2} + 7p_{1}^{2}), Â_{3} = \frac{4}{3^{2}·5·7}(−16p_{3} + 44p_{2}p_{1} − 31p_{1}^{3}),
ph_{0} = 0, ph_{1} = p_{1}, ph_{2} = \frac{1}{24}(−4p_{2} + p_{1}62), ph_{3} = \frac{1}{2^{7}·3^{3}·5}(144p_{3} − 28p_{2}p_{1} + p_{1}^{3}).

Now following Stolz we define

S_{k}(W) = \frac{1}{8}L_{k}(W) + \frac{|bP_{4k}|}{a_{k}} (c_{k}Â_{k}(W) + (−1)^{k}d_{k}(Â(W)ph(W))_{k})

where a_{k} = 1 for k even and 2 for k odd, and c_{k} and d_{k} are integers such that

c_{k}num(B_{k}/4k) + d_{k}denom(B_{k}/4k) = 1.

The symbol B_{k} denotes the kth Bernoulli number and num(B_{k}/4k) and denom(B_{k}/4k) denote respectively the numerator and denominator of the fraction B_{k}/4k when expressed in lowest terms. As M is a rational homology sphere H^{∗}(W; ℚ) ≅ H^{∗}(W, M; ℚ) and we use this isomorphism to view each Pontryagin class p_{i}(W) as an element of H^{∗}(W, M; ℚ). Hence we may evaluate S_{k}(W) against on the fundamental class of W, [W, ∂W] ∈ H_{4k}(W), to obtain the rational number < S_{k}(W), [W, ∂W] >.

Theorem 2.55 (c.f. [St] Theorem). Let M be a (4k − 1)-dimensional spin rational homotopy sphere bounding a spin manifold W and let σ(W) denote the signature of W. Then the coefficient of p_{k}(W) in S_{k}(W) is zero and

s_{1}(M) := \frac{1}{|bP_{4k}|} (< S_{k}(W), [W, ∂W] > −\frac{1}{8}σ(W)) mod ℤ

is a well defined smooth invariant of M. If M = Σ is a homotopy sphere then

−|bP_{4k}| · s_{1}(Σ) = s(Σ) ∈ ℤ/|bP_{4k}|

computes Brumfiel’s splitting invariant of Σ.

Proof. The first and third assertions of the Theorem are proven by Stolz. Moreover, Stolz’s proof that the quantity \frac{1}{8}σ(W)− < S_{k}(W), [W, ∂W > (mod|bP_{4k}|) is

12. THE INVARIANTS s_{1}, s̄_{1} AND μ 67

independent of the choice of spin coboundary W makes no essential use of the fact that M is a homotopy sphere as opposed to a rational homology sphere. We present this proof again to reassure the reader of this claim. Suppose that W and W′ are two spin manifolds bounding M. Let X be the closed spin manifold obtained by gluing W to −W′ along M and let i : W ↪ X and i′ : −W′ ↪ X denote the inclusions. As M is a rational homology sphere, i^{∗} ⊕ i^{′∗} : H^{2k}(X; ℚ) → H^{2k}(W; ℚ) ⊕ H^{2k}(W′ℚ) is an isomorphism and (i^{∗} ⊕ i^{′∗})p_{i}(X) = (p_{i}(W), p_{i}(−W′)) for i < k. It follows that

σ(X) = σ(W) − σ(W′)

and

< S_{k}(X), [X] >=< S_{k}(W), [W, ∂W] > − < S_{k}(W′), [W′, ∂W′] > .

We now calculate that

(1/|bP_{4k}|) (< S_{k}(W), [W, ∂W] > −\frac{1}{8}σ(W))

−(1/|bP_{4k}|) (< S_{k}(W′), [W′, ∂W′] > −\frac{1}{8}σ(W′))
= (1/|bP_{4k}|) (< S_{k}(X), [W, ∂X] > \frac{1}{8}σ(X))
= (1/|bP_{4k}|)(L_{k}(X) − \frac{1}{8}σ(X))
+ (c_{k}(1/a_{k}) < Â(X), [X] > +(−1)^{k}d_{k}(1/a_{k}) < (Â(X)ph(X))_{k}, [X] >)
= 0 mod ℤ

because σ(X) =< L_{k}(X), [X] > by Hirzebruch’s signature theorem the quantities (1/a_{k}) < Â(X), [X] > and (1/a_{k}) < (Â(X)ph(X))_{k}, [X] > are integers by the Hirzebruch-Riemann-Roch Theorem.

Remark 2.56. We mentioned above that Stolz’s s_{1}-invariant generalizes the Eels-Kuiper μ-invariant and we now explain this remark. For W and M as before Eels and Kuiper made the following definitions:

N_{k}(W) := \frac{1}{8 · |bP_{4k}|}L_{k}(W) + \frac{1}{a_{k}} Â_{k}(W)

12. THE INVARIANTS s_{1}, s̄_{1} AND μ 68

and

μ(M) :=< N_{k}(W), [W, ∂W] > −\frac{1}{8 · |bP_{4k}|}σ(W).

Now when num(B_{4k}/4k) = 1 (which is the case for k = 2 and when k = 4) then we may take c_{k} = 1 and d_{k} = 0 in the definition of S_{k}(W) and s_{1}(M) and then s_{1}(M) = μ(M). However, when num(B_{4k}/4k) ≠ 1 the s_{1}-invariant is strictly shaper than the μ-invariant.

Observe that for two closed spin (4k − 1)-manifolds M_{0} and M_{1} bounding spin manifolds W_{0} and W_{1} that M_{0}#M_{1} = ∂(W_{0}♮W_{1}) and hence s_{1}(M_{0}#M_{1}) = s_{1}(M_{0}) + s_{1}(M_{1}). Observe also that

|bP_{4k}| = max{denom(s_{1}(Σ))|Σ ∈ Θ_{4k−1}}

and then define

s̄_{1}(M) := |bP_{4j}|s_{1}(M) ∈ ℚ/ℤ.

It follows immediately that s̄_{1}(Σ) = 0 for any homotopy sphere and therefore that s̄_{1} is an almost diffeomorphism invariant. In fact, when k = 2 Kreck and Stolz show that s̄_{1} is a topological invariant [KS]. We shall later identify the s̄_{1}-invariant of a highly connected rational homology sphere in dimensions 7 and 15 with the Kervaire-Arf invariant of its quadratic linking function. To conclude this section we present the calculations of s_{1} for 2-connected 7-manifolds and 6-connected 15-manifolds. If L is an 8j-dimensional handlebody with tangential invariant α then the Pontryagin classes of L are as follows:

p_{i}(L) = \{ \begin{matrix}(2j − 1))!a_{k}α & i = j \\ 0 & i ≠ j \end{matrix} \} .

It follows that the only possibly non-zero term in S_{2j}(L) is that containing p_{j}(L)^{2}. Now let L_{(j,2j)} and Â_{(j,2j)} denote the coefficient of p_{j}^{2} in respectively L_{2j} and Â_{2j}. We also need to know the coefficients of p_{j} in ph(L)_{j} and p_{j}^{2} in ph(L)_{2j} but these may

12. THE INVARIANTS s_{1}, s̄_{1} AND μ 69

be computed using Newton’s formula ([MS] Ex 16.A) and the fact that all but one Pontryagin class vanish. We calculate that

S_{2j}(L) = S_{(j,2j)}(L)p_{j}^{2} = S̃_{(j,2j)}(L)α^{2}

where S̃_{(j,2j)}(L) = ((2j − 1)!)^{2}S_{(j,2j)}(L) and

S_{(j,2j)}(L) := \frac{1}{8}L_{(j,2j)} + |bP_{8j}| (c_{2j}Â_{(j,2j)} + d_{2j} (Â_{(j,2j)} \frac{(−1)^{j}}{2j!(2j − 1)!} + \frac{1}{(2j − 1)!})) .

Proposition 2.57. Let P be a highly connected manifold of dimension 7 or 15 and let P #Σ_{P} = ∂L where the handlebody L has quadratic function κ(L) = κ(H, λ, α). Then

dim(P) = 8j − 1 : s_{1}(P) = (1/|bP_{8j}|) (S̃_{(j,2j)}λ^{−1}(α, α) − \frac{1}{8}σ(λ))) ,

\begin{matrix}dim(P) = 7 : & s_{1}(P) = \frac{1}{28} · \frac{1}{8}(λ^{−1}(α, α) − σ(λ)), & s̄_{1}(P) = \frac{1}{8}(λ^{−1}(α, α) − σ(λ)), \\ dim(P) = 15 : & s_{1}(P) = \frac{1}{8,128} · \frac{1}{8}(λ^{−1}(α, α) − σ(λ)), & s̄_{1}(P) = \frac{1}{8}(λ^{−1}(α, α) − σ(λ)). \end{matrix}

Proof. By definition s(Σ_{P}) = 0. Thus s_{1}(Σ_{P}) = 0 and s_{1}(P) = s_{1}(∂L). The term λ^{−1}(α, α) appears since to compute < α^{2}, [L, ∂L] > we must transport α ∈ H^{4j}(L; ℚ) to λ̂^{−1}(α) ∈ H^{4j}(L, ∂L; ℚ) and λ̂^{−1}(α)^{2} = λ^{−1}(α, α). The specific computations for dimensions 7 and 15 come from from knowledge of L_{(1,2)}, L_{(2,4)}, Â_{(1,2)}, Â_{(2,4)}, B_{2} and B_{4} which may all be found in [Hi]. Alternatively, since B_{2} = B_{4} = 1/30, s_{1} = μ in dimensions 7 and 15 and these calculations are in [EK].

CHAPTER 3 Stable algebra of quadratic functions

In this chapter we establish the algebraic results we shall need to classify the manifolds P up to almost diffeomorphism. In Chapter 4 we shall apply these results. Recall that two quadratic functions κ_{0} and κ_{1} belonging to a category F^{∗} are stably F^{∗}-equivalent if there are nonsingular quadratic functions μ_{0} and μ_{1} also belonging to F^{∗} such that κ_{0} ⊕ μ_{0} ≅ κ_{1} ⊕ μ_{1}. In this chapter we establish that stable F^{∗}- equivalence is closely related to two further relations between quadratic functions. The first of these is that of isometry between their induced quadratic linking families and the second is orthogonality which we shall define presently.

Definition 3.1. An isometric embedding i_{0} : κ_{0}(H_{0}) ↪ κ(H) between nondegenerate quadratic functions is said to be primitive if i_{0}(H_{0}) is a summand of H. The orthogonal complement of a primitive embedding i_{0}, denoted by i_{0}(κ_{0})^{⊥}, is defined to be κ|_{i_{0}(H_{0})^{⊥}} where

i_{0}(H_{0})^{⊥} := {v ∈ H|λ(v, i_{0}(v_{0})) = 0 ∀v_{0} ∈ H_{0}}.

Example 3.2. Let κ = κ(H, λ, 0) and κ_{0} = κ(H_{0}, λ, 0) be quadratic forms defined on H = ℤ^{2}[v, w] and H_{0} = ℤ[w_{0}], by the following equations: λ(v, v) = 0, λ(v, w) = 1, λ(w, w) = n and λ_{0}(w_{0}, w_{0}) = n, where n ∈ ℤ − {−1, 0, 1}. Then i_{0} : H_{0} ↪ H, w_{0} ↦ w, is a primitive embedding of a nondegenerate but singular form into a nonsingular form. In this case i_{0}(H_{0})^{⊥} = ℤ[nv − w] and i_{0}(κ_{0})^{⊥} is defined by

i_{0}(κ_{0})^{⊥}(nv − w) = n.

3. STABLE ALGEBRA OF QUADRATIC FUNCTIONS 71

Definition 3.3. Let κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}) be quadratic functions belonging to the category F^{∗}.

1. If κ_{0} and κ_{1} are nondegenerate then they are F^{∗}-orthogonal if there exists a nonsingular quadratic function κ belonging to F^{∗} and primitive embeddings i_{0} : κ_{0} ↪ κ and i_{1} : κ_{1} ↪ κ such that i_{0}(κ_{0})^{⊥} = i_{1}(κ_{1}).

2. In general, κ_{0} and κ_{1} are F^{∗}-orthogonal if there exits splittings (see Section 9)

κ_{0} = α_{0}|_{F_{0}} ⊕ κ_{0}(Ψ_{0}) and κ_{1} = α_{1}|_{F_{1}} ⊕ κ_{1}(Ψ_{1}) such that α_{0}|_{F_{0}} ≅ α_{1}|_{F_{1}} and such that κ_{0}(Ψ_{0}) is F^{∗}-orthogonal to κ_{1}(Ψ_{1}).

In both cases we write κ_{0} ⊥_{F^{∗}} κ_{1}.

We now state the theorem whose proof is the subject of the chapter.

Theorem 3.4. Let κ_{0} and κ_{1} be two quadratic functions in the same family F^{∗}. Then the following are equivalent.

1. κ_{0} is stably F^{∗}-equivalent to κ_{1}, κ_{0} ∼_{F^{∗}} κ_{1}.

2. The quadratic linking families induced by κ_{0} and κ_{1} are isometric, δ^{∗}(κ_{0}) ≅ δ^{∗}(κ_{1}).

3. κ_{0} and κ_{1}^{−} are F^{∗}-orthogonal, κ_{0} ⊥_{F^{∗}} κ_{1}^{−}.

Remark 3.5. In the nondegenerate case when F^{∗} = F^{o}, the equivalence of 1 and 2 was proven by Durfee [Du1], and Wall [Wa4]. Though he did not state it this way, Wilkens [Wi1] proved the equivalence of 1 and 2 for F^{c} in the case where the induced quadratic linking family is indecomposable. Nikulin proved the equivalence of 2 and 3 ([Ni] Corollary 1.6.2) in the nondegenerate case where F^{∗} = F^{o}. Of course, the family F^{c} is of primary concern for the classification of manifolds in dimensions 7 and 15, however the proofs given below may shed some light on the results when F^{∗} = F^{o} or F^{ev}. In particular, Durfee’s methods use quite delicate p-adic techniques, Wall’s

1. GLUING AND SPLITTING NONDEGENERATE QUADRATIC FUNCTIONS 72

proof is somewhat opaque (to this reader) though similar to our own and Nikulin’s proofs are extremely brief. Our perspective on these results is that they are algebraic counterparts to the topology of handlebodies and their boundaries.

1. Gluing and splitting nondegenerate quadratic functions

The gluing lemma which follows is inspired by the following topological idea: suppose that f : P_{0} → P_{1} is an almost diffeomorphism of highly connected rational homology spheres with coboundaries L_{0} and L_{1}. We could then form the almost smooth manifold M = L_{0} ∪_{f} −L_{1} which would be a highly connected 8j-manifold and there would be a Mayer-Vietoris decomposition for H_{4j}(W) relating the domains of the quadratic functions of M, L_{0} and L_{1}.

[figure: 0; H_{4j}(L_{0}) ⊕ H_{4j}(−L_{1}); H_{4j}(M); H_{4j−1}(P); 0; 0; H_{0} ⊕ H_{1}; H; T G; 0]

The idea then is to manufacture the algebraic data for a manifold M from the algebraic data corresponding to the almost diffeomorphism f. If κ_{0} = κ(L_{0}) and κ_{1} = κ(L_{1}) are the quadratic functions of the handlebodies L_{0} and L_{1} then this data is an isomorphism θ : δ^{∗}(L_{0}) ≅ δ^{∗}(L_{1}) of the induced quadratic linking families. In the following lemma we show how to define a quadratic function in the class F^{∗} from θ, κ_{0} and κ_{1} which we shall denote by κ_{0} ∪_{θ} κ_{1}^{−}. All of the notation follows the conventions established in Chapter 2. In particular the fundamental sequence of a non-degenerate quadratic function κ_{i} = κ(H_{i}, λ_{i}, α_{i}) is

0 ⟶ H_{i} ⟶^{λ̂_{i}} H_{i}^{∗} ⟶^{π_{i}} T G_{i} ⟶ 0

and for x_{i} ∈ H_{i}^{∗} we shall write [x_{i}] for π_{i}(x_{i}).

Lemma 3.6 (Gluing Lemma). Let F^{∗} be one of the families F^{c}, F^{ev} or F^{o} and let κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}) be nondegenerate quadratic functions

1. GLUING AND SPLITTING NONDEGENERATE QUADRATIC FUNCTIONS 73

in the same family F^{∗}. Suppose that θ : Cok(λ̂_{0}) → Cok(λ̂_{1}) is an isometry of the induced quadratic linking families;

δ^{∗}(κ_{0}) ⟶^{θ} δ^{∗}(κ_{1}).

Let H ⊂ H_{0}^{∗} ⊕ H_{1}^{∗} be the set {(x_{0}, x_{1}) : θ([x_{0}]) = [x_{1}])} and λ the function

\begin{aligned}λ : & H × H & → & ℚ \\ & [(x_{0}, x_{1}), (y_{0}, y_{1})] & ↦ & λ_{0}^{−1}(x_{0}, y_{0}) − λ_{1}^{−1}(x_{1}, y_{1}). \end{aligned}

Then 1. λ is a nonsingular integral bilinear form on the free abelian group H,

2. there is a canonical isometric embedding (λ_{0} ⊕ −λ_{1}) ↪ λ,

3. the element (α_{0}, α_{1}) ∈ H,

4. if α := λ̂((α_{0}, α_{1})), then the quadratic function κ_{0} ∪_{θ} κ_{1}^{−} := κ(H, λ, α) is of type F^{∗},

5. there is a canonical isometric embedding κ_{0} ⊕ κ_{1}^{−} ↪ κ_{0} ∪_{θ} κ_{1}^{−}.

Proof. (1. & 2.) Observe that H is the kernel of the homomorphism H_{0}^{∗}⊕H_{1}^{∗} → T G_{1} which takes (x_{0}, x_{1}) to θ([x_{0}]) − [x_{1}]. Thus H is a subgroup of the free abelian group H_{0}^{∗} ⊕ H_{1}^{∗} so is itself a free abelian group. Now let (x_{0}, x_{1}) and (y_{0}, y_{1}) ∈ H. We show first that λ[(x_{0}, x_{1}), (y_{0}, y_{1})] is an integer.

λ[(x_{0}, x_{1}), (y_{0}, y_{1})] mod ℤ = λ_{0}^{−1}(x_{0}, y_{0}) − λ_{1}^{−1}(x_{1}, y_{1})  mod ℤ
= b_{0}([x_{0}], [y_{0}]) − b_{1}([x_{1}], [y_{1}])  mod ℤ
= b_{0}([x_{0}], [y_{0}]) − b_{1}(θ([x_{0}]), θ([y_{0}])) mod ℤ
= 0  mod ℤ

The last equality holds as q^{∗}(κ_{0}) = θ^{∗}q^{∗}(κ_{1}) entails that b_{0} = b_{1} ∘ (θ × θ). We next show that λ is nonsingular. Consider first the inclusions

\begin{aligned}i_{0} : & H_{0} & ↪ & H \\ & v_{0} & ↦ & (λ̂_{0}(v_{0}), 0) \end{aligned}  \begin{aligned}i_{1} : & H_{1} & ↪ & H \\ & v_{1} & ↦ & (0, −λ̂_{1}(v_{1})). \end{aligned}

1. GLUING AND SPLITTING NONDEGENERATE QUADRATIC FUNCTIONS 74

The sum of i_{0} and i_{1}, i_{0} ⊕ i_{1} : H_{0} ⊕ H_{1} ↪ H defines an embedding (λ_{0} ⊕ −λ_{1}) ↪ λ for we calculate that

λ[(λ̂_{0}(v_{0}), −λ̂_{1}(v_{1})), (λ̂_{0}(w_{0}), −λ̂_{1}(w_{1}))] = λ_{0}^{−1}(λ̂_{0}(v_{0}), λ̂_{0}(w_{0}))
−λ_{1}^{−1}(−λ̂_{1}(v_{1}), −λ̂_{1}(w_{1}))
= λ_{0}(v_{0}, w_{0}) − λ_{1}(v_{1}, w_{1})
= (λ_{0} ⊕ −λ_{1})[(v_{0}, w_{0}), (v_{1}, w_{1})].

We now define T G to be the subgroup of T G_{0}⊕T G_{1} consisting of pairs ([x_{0}], θ[x_{0}]) and we let π : H → T G be the map (x_{0}, x_{1}) → ([x_{0}], [x_{1}]). The linking form b := b(λ_{0}) ⊕ b(−λ_{1}) is defined on T G_{0} ⊕ T G_{1} with adjoint map b̂: T G_{0} ⊕ T G_{1} → (T G_{0} ⊕ T G_{1})^{∧}. Setting R_{T G} : (T G_{0} ⊕ T G_{1})^{∧} → T G^{∧} to be the restriction map we define Π : H_{0}^{∗} ⊕ H_{1}^{∗} → T G^{∧} to be the composition R_{T G} ∘ b̂ ∘ π_{0} ⊕ π_{1}. It is clear that i(H) ⊂ Ker(Π) where i : H → H_{0}^{∗} ⊕ H_{1}^{∗} is the inclusion i(x_{0}, x_{1}) = (x_{0}, x_{1}). Moreover, Π|_{T G_{0}⊕0} is onto and since the groups involved are finite we conclude that Ker(Π) = i(H). We shall show that the following diagram is commutative with exact rows and columns (where we have identified (H_{0} ⊕ H_{1}) = (H_{0} ⊕ H_{1})^{∗∗}.)

Diagram 3.7.

[figure: 0; 0; 0; H_{0} ⊕ H_{1}; H; T G; 0; 0; H^{∗}; H_{0}^{∗} ⊕ H_{1}^{∗}; T G^{∧}; 0; T G^{∧∧}; T G^{∧}; 0; 0; i_{0}⊕i_{1}; i^{∗}; π; i; λ̂; i_{0}^{∗}⊕i_{1}^{∗}; Π̂; π̂; Π]

1. GLUING AND SPLITTING NONDEGENERATE QUADRATIC FUNCTIONS 75

It is evident that i_{0} ⊕i_{1} is injective, that Im(i_{0} ⊕i_{1}) = Ker(π) and that π is onto. It is similarly clear that i is injective and we have already observed that Π is onto and that Ker(Π) = Im(i). Hence the first row and second column are exact. The exactness of first column and second row follow from the duality we reviewed in section 7. We must show that i^{∗} : (H_{0} ⊕ H_{1})^{∗∗} → H^{∗} is the same as λ̂ ∘ (i_{0} ⊕ i_{1}). For v_{i} ∈ H_{i} let v_{i}^{∗∗} ∈ H_{i}^{∗∗} denote the element ϕ ↦ ϕ(v_{i}) for ϕ ∈ H_{i}^{∗}. We compute that for (x_{0}, y_{0}) ∈ H

i^{∗}(v_{0}^{∗∗}, v_{1}^{∗∗})[(x_{0}, x_{1})] = (v_{0}^{∗∗}, v_{1}^{∗∗})[(x_{0}, x_{1})]
= x_{0}(v_{0}) + x_{1}(v_{1})
= λ_{0}^{−1}(λ̂(v_{0}), x_{0}) − λ_{1}^{−1}(−λ̂_{1}(v_{1}), x_{1})
= λ[(λ̂(v_{0}), −λ̂(v_{1})), (x_{0}, x_{1})]
= λ̂[(i_{0} ⊕ i_{1})(v_{0}, v_{1})][(x_{0}, x_{1})]

and thus i^{∗} = λ̂ ∘ (i_{0} ⊕ i_{1}) after we have identified H_{i} = H_{i}^{∗∗} by v_{i} ↦ v_{i}^{∗∗}. A similar computation shows that i = (i_{0}^{∗} ⊕ i_{1}^{∗}) ∘ λ̂. We now require the following lemma whose proof is left to the reader.

Lemma 3.8. Let A →^{f} B →^{g} C be the composition of a pair of injective homomorphisms of abelian groups. Then there is a short exact sequence

0 → Cok(f) → Cok(g ∘ f) → Cok(g) → 0.

If we apply the lemma to λ̂ ∘ (i_{0} ⊕ i_{1}) we see that

0 → T G^{∧∧} → T G → Cok(λ̂) → 0

is exact, that Cok(λ̂) = 0 and therefore that λ is nonsingular. (3.) By the definition of an isometry of a quadratic linking family θ([α_{0}]) = [α_{1}] and hence (α_{0}, α_{1}) ∈ H. (4.) Suppose that F^{∗} = F^{c}. We must show that α = λ̂((α_{0}, α_{1})) is characteristic for

1. GLUING AND SPLITTING NONDEGENERATE QUADRATIC FUNCTIONS 76

λ given that each α_{i} is characteristic for λ_{i}. Let x = (x_{0}, x_{1}) ∈ H and observe that

[λ(x, x) + α(x)]/2 mod ℤ = [λ(x, x) + λ̂(α_{0}, α_{1})(x)]/2  mod ℤ
= [λ_{0}^{−1}(x_{0}, x_{0} + α_{0}) − λ_{1}^{−1}(x_{1}, x_{1} + α_{1})]/2 mod ℤ
= q^{c}(κ_{0})([x_{0}]) − q^{c}(κ_{1})(θ([x_{0}]))  mod ℤ
= 0  mod ℤ.

Thus λ̂(α) is characteristic for λ. If F^{∗} = F^{ev} then we must show that λ is even. We again let x = (x_{0}, x_{1}) ∈ H and compute

λ(x, x)/2 mod ℤ = [λ_{0}^{−1}(x_{0}, x_{0}) − λ_{1}^{−1}(x_{1}, x_{1})]/2  mod ℤ
= q^{ev}(κ_{0})([x_{0}]) − q^{ev}(κ_{1})(θ([x_{0}])) mod ℤ
= 0  mod ℤ.

Thus λ is even. If F^{∗} = F^{o} then we must show that λ is even and that α = 0. However, α = λ̂((α_{0}, α_{1})) = λ̂((0, 0)) = 0 and λ is even by the computation for the case F^{∗} = F^{ev}. (5.) We have already proven this result for the quadratic forms (λ_{0} ⊕ −λ_{1}) ↪ λ so we need only show that the inclusions i_{0} and i_{1} also commute with respect to α_{0}, α_{1} and α := λ̂((α_{0}, −α_{1})). We do this now for (v_{0}, v_{1}) ∈ H_{0} ⊕ H_{1}.

α[(i_{0} ⊕ i_{1})(v_{0}, v_{1})] = λ[(α_{0}, α_{1}), (λ̂_{0}(v_{0}), −λ̂_{1}(v_{1}))]
= λ̂_{0}^{−1}(α_{0}, λ̂_{0}(v_{0})) − λ̂_{1}^{−1}(α_{1}, −λ̂_{1}(v_{1}))
= α_{0}(v_{0}) + α_{1}(v_{1})

Remark 3.9. When κ_{0} = κ_{1} and θ = Id then we shall simply write κ_{0} ∪ κ_{0}^{−} for the “double” of κ.

The Splitting Lemma which follows shows that every primitive embedding i_{0} : κ_{0} ↪ κ of a nondegenerate quadratic function into a nonsingular quadratic function gives rise to a decomposition of κ as in the Gluing Lemma: κ ≅ i_{0}(κ_{0}) ∪_{θ} i_{0}(κ_{0})^{⊥−}.

1. GLUING AND SPLITTING NONDEGENERATE QUADRATIC FUNCTIONS 77

It is topologically motivated by considering the embedding of a handlebody L_{0} with boundary a rational homology sphere into a closed highly connected 8j-manifold L. The complement of this embedding is itself a handlebody with boundary the same rational homology sphere (though the orientation is reversed) and L is obtained by gluing L_{0} to (L − L_{0}) along their common boundary.

Lemma 3.10 (Splitting Lemma). Let κ = κ(H, λ, α) be a nonsingular quadratic function in the family F^{∗} and let H_{0} be a summand of H such that κ_{0} := κ|_{H_{0}} is nondegenerate. Then

1. the group H_{1} := H_{0}^{⊥} = {v ∈ H : λ(v, v_{0}) = 0 ∀v_{0} ∈ H_{0}} is a summand of H,

2. the quadratic functions κ_{1} := (κ|_{H_{1}})^{−} and κ_{0} are nondegenerate members of F^{∗},

3. there is an isomorphism θ_{κ} : δ^{∗}(κ_{0}) ≅ δ^{∗}(κ_{1}),

4. κ and κ_{0} ∪_{θ_{κ}} κ_{1}^{−} are isometric.

Proof. (1.) If rv belongs to H_{1} for a nonzero integer r then rv pairs trivially with all of H_{0} but this entails that v does also. Thus v ∈ H_{1} and so H_{1} is a summand. (2.) By definition a quadratic function is nondegenerate if and only if the associated bilinear form is nondegenerate so we consider the bilinear forms λ, λ_{0} := λ|_{H_{0}×H_{0}} and λ_{1} := −λ|_{H_{1}×H_{1}}. We let a subscript ℚ denote the forms obtained after tensoring with the rationals so that for example λ_{ℚ} : (H ⊗ ℚ) × (H ⊗ ℚ) → ℚ. As λ_{0ℚ} is a nonsingular subform of the nonsingular form λ_{ℚ} it follows that λ_{ℚ} = λ_{0ℚ} ⊕ −λ_{1ℚ} and therefore that λ_{1ℚ} is nonsingular. Thus λ_{1} and κ_{1} are nondegenerate. Now let α_{i} = α|_{H_{i}}, i = 0, 1. That both κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}) belong to F^{∗} follows from the fact that the properties of being homogeneous (F^{o}), having an even quadratic form (F^{ev}) and being even (F^{c}) are all preserved upon restricting to a summand of the domain of a quadratic function.

1. GLUING AND SPLITTING NONDEGENERATE QUADRATIC FUNCTIONS 78

(3.) We shall present the proof in the case where F^{∗} = F^{c} as the other cases are similar and easier. From the fact that λ_{ℚ} = λ_{0ℚ} ⊕ −λ_{1ℚ}, we deduce that the rational vector space H ⊗ ℚ = (H_{0} ⊕ H_{1}) ⊗ ℚ and thus H_{0} ⊕ H_{1} is a subgroup of H with finite index. We therefore define π to be the projection from H onto this finite group T G := H/(H_{0} ⊕ H_{1}). We now define a series of groups and homomorphisms which shall fit into an extensive commutative diagram. Let π′ be the projection from H^{∗} to the finite group λ̂(T G) := H^{∗}/λ̂(H_{0} ⊕ H_{1})) which is canonically isomorphic to T G via the map induced by λ̂. Let i_{0} : H_{0} ↪ H and i_{1} : H_{1} ↪ H denote the obvious inclusion maps and as usual let π_{i} denote the projection from H_{i}^{∗} to H_{i}^{∗}/λ̂(H_{i}) = T G_{i} for i = 0, 1. In addition define J : H → H_{0}^{∗} ⊕ H_{1}^{∗} to be the map which sends v ∈ H to (λ̂(v)|_{H_{0}}, λ̂(v)|_{H_{1}}), I to be the map which sends π(v) to (π_{0} ⊕ π_{1})(J(v)) and K : λ̂(T G) → T G_{0} ⊕ T G_{1} be the map which sends π′(x) to the pair (π_{0}(i_{0}^{∗}(x)), π_{1}(i_{1}^{∗}(x))). That I and K are well defined follows from standard diagram chases in Diagram 3.11. Note that Diagram 3.11 corresponds to Diagram 3.7 of Lemma 3.6 and extends that diagram. It is commutative and all sets of collinear arrows represent exact sequences of homomorphisms.

Diagram 3.11.

[figure: 0; 0; 0; 0; 0; H_{0} ⊕ H_{1}; H; T G; 0; 0; H^{∗}; H_{0}^{∗} ⊕ H_{1}^{∗}; T G^{∧}; 0; 0; λ̂(T G); T G_{0} ⊕ T G_{1}; T G^{∧}; 0; 0; λ̂(T G)^{∧}; λ̂(T G)^{∧}; 0; 0; 0; i_{0}⊕i_{1}; λ̂∘(i_{0}⊕i_{1}); λ̂_{0}⊕−λ̂_{1}; π; J; I; λ̂; i_{0}^{∗}⊕i_{1}^{∗}; π′; π^{∧}; π_{0}⊕π_{1}; Id; K; Id]

Let pr_{i} : T G_{0} ⊕ T G_{1} → T G_{i} is the projection onto the indicated factor and for v ∈ H suppose that pr_{0}(I(π(v))) = 0. By definition this means that there is some v_{0} ∈ H_{0} such that λ̂(v)|_{H_{0}} = λ̂_{0}(v_{0}). That is,

λ(v_{0}, w) = λ(v, w) ∀w ∈ H_{0}.

It follows that λ(v − v_{0}, w) = 0 for all w ∈ H_{0} and hence that (v − v_{0}) ∈ H_{0}^{⊥} = H_{1}. Therefore v = v_{0} + (v − v_{0}) = (i_{0} ⊕ i_{1})(v_{0}, v − v_{0}) and hence π(v) = 0 ∈ T G. Hence pr_{0} ∘ I is injective and a completely similar argument shows the same for pr_{1} ∘ I. Now by Lemma 3.8 applied to λ̂_{0} ⊕ −λ̂_{1} = J ∘ (i_{0} ⊕ i_{1}) we see that there is a short exact sequence

0 → T G → T G_{0} ⊕ T G_{1} → λ̂(T G)^{∧} → 0.

Now |λ̂(T G)^{∧}| = |T G| and so |T G|^{2} = |T G_{0}| · |T G_{1}|. But since pr_{i} ∘ I is injective |T G_{i}| ≥ |T G| and so |T G| = |T G_{i}|. We conclude that that pr_{i} ∘ I is an isomorphism.

We show next that δ^{c}(κ_{0}) ⊕ −δ^{c}(κ_{1}) is identically zero on I(T G) ⊂ T G_{0} ⊕ T G_{1}. Given v ∈ H there is a non-zero integer r and elements v_{i} ∈ H_{i} for i = 0, 1 such that v = r(v_{0} + v_{1}). Thus, for any w ∈ H_{0}, λ(v, w) = λ(rv_{0}, w) = rλ_{0}(v_{0}, w) and hence λ̂(v)|_{H_{0}} = rλ̂_{0}(v_{0}). Where as, for any w ∈ H_{1}, λ(v, w) = λ(rv_{1}, w) = −rλ_{1}(v_{1}, w) and λ̂(v)|_{H_{1}} = rλ̂_{1}(−v_{1}). We now calculate

(δ^{c}(κ_{0}) ⊕ −δ^{c}(κ_{1}))[I(π(v))] = [λ̂_{0}^{−1}(λ̂(v)|_{H_{0}}, λ̂(v)|_{H_{0}} + α_{0})
−λ̂_{1}^{−1}(λ̂(v)|_{H_{1}}, λ̂(v)|_{H_{1}} + α_{1})]/2
= [λ(v, v_{0}) + α_{0}(v_{0}) − λ(v, −v_{1}) + α_{1}(−v_{1})]/2r
= [λ(v, v_{0} + v_{1}) + α(v_{0} + v_{1})]/2r
= [λ(v, v) + α(v)]/2.

Now this last quantity is integral as κ ∈ F^{c} and α is characteristic for λ. Thus

(δ^{c}(κ_{0}) ⊕ −δ^{c}(κ_{1}))[I(π(v))] = 0 ∈ ℚ/ℤ

for every π(v) ∈ T G. So it follows that

θ_{κ} := (I ∘ pr_{1}) ∘ (I ∘ pr_{0})^{−1} : T G_{0} → T G_{1}

is an isometry from δ^{c}(κ_{0}) to δ^{c}(κ_{1}). In fact I(T G) = {([x_{0}], θ_{κ}([x_{0}]))|[x_{0}] ∈ T G_{0}} ⊂ T G_{0} ⊕ T G_{1} and since δ^{c}(κ_{0}) ⊕ −δ^{c}(κ_{1}) vanishes identically on I(T G) we see that for all [x_{0}] ∈ T G_{0}

δ^{c}(κ_{0})([x_{0}]) = δ^{c}(κ_{1})(θ_{κ}([x_{0}])).

(4.) Observe that J(H) = {(x_{0}, x_{1}) ∈ H_{0}^{∗}⊕H_{1}^{∗}|θ_{κ}([x_{0}]) = [x_{1}]} which is the underlying abelian group of κ_{0}∪_{θ_{κ}} κ_{1}^{−}. As above, for any v ∈ H we obtain a non-zero integer r and v_{i} ∈ H_{i}, i = 0, 1, such that v = r(v_{0}+v_{1}), λ̂(v)|_{H_{0}} = rλ̂_{0}(v_{0}) and rλ̂(v)|_{H_{1}} = rλ̂_{1}(−v_{1}).

2. STABLE EQUIVALENCE OF QUADRATIC FUNCTIONS 81

A similar calculation to preceding one yields

(κ_{0} ∪_{θ_{κ}} κ_{1}^{−})(J(v)) = (κ_{0} ∪_{θ_{κ}} κ_{1}^{−})((λ̂(v)|_{H_{0}}, λ̂(v)|_{H_{1}}))
= λ̂_{0}^{−1}(λ̂(v)|_{H_{0}}, λ̂(v)|_{H_{0}} + α_{0}) − λ̂_{1}^{−1}(λ̂(v)|_{H_{1}}, λ̂(v)|_{H_{1}} + α_{1})
= [λ(v_{0}, v) + α_{0}(v_{0})]/r − [λ(−v_{1}, v) + α(−v_{1})]/r
= [λ(v_{0}, v) + α(v_{0}) + λ(v_{1}, v) + α(v_{1})]/r
= λ(v_{0} + v_{1}, v)/r + α(v_{0} + v_{1})/r
= λ(v, v) + α(v)
= κ(v).

So J is an isometry from κ to κ_{0} ∪_{θ_{κ}} κ_{1}^{−}.

2. Stable equivalence of quadratic functions

In this section we shall establish that the induced family of quadratic linking functions is a complete stable invariant of quadratic functions. Our method of proof is inspired by Wilkens’ proof of the analogous topological result (see Section 4.1).

Lemma 3.12. Let κ_{0}(H_{0}, λ_{0}, α_{0}) and κ(H_{1}, λ_{1}, α_{1}) be nondegenerate quadratic functions in the family F^{∗}. Suppose that θ is an isometry of the induced quadratic linking families, θ : δ^{∗}(κ_{0}) ≅ δ^{∗}(κ_{1}). Then, there exists a nonsingular quadratic function μ_{0} and an isometry Θ : κ_{0} ⊕ μ_{0} ≅ κ_{1} ⊕ (κ_{0} ∪_{θ} κ_{1}^{−}) such that δ^{∗}(Θ) = θ.

Proof. Consider the four isometric embeddings:

\begin{aligned}j_{0} : & κ_{0} & ↪ & (κ_{0} ∪ κ_{0}^{−}) \\ & v & ↦ & (λ̂_{0}(v), 0) \end{aligned}  \begin{aligned}j_{1} : & κ_{0}^{−} & ↪ & (κ_{0} ∪ κ_{0}^{−}) \\ & v & ↦ & (0, −λ̂_{0}(v)) \end{aligned}
\begin{aligned}i_{0} : & κ_{0} & ↪ & (κ_{0} ∪_{θ} κ_{1}^{−}) \\ & v & ↦ & (λ̂_{0}(v), 0) \end{aligned}  \begin{aligned}i_{1} : & κ_{1}^{−} & ↪ & (κ_{0} ∪_{θ} κ_{1}^{−}) \\ & v & ↦ & (0, −λ̂_{1}(v)). \end{aligned}

We shall prove that there is a further isometric embedding I : (κ_{0}∪κ_{0}^{−}) ↪ (κ_{0}∪_{θ}κ_{1}^{−})⊕ (κ_{0} ∪_{θ} κ_{1}^{−})^{−}, a nonsingular quadratic function μ_{0} and an isometry Φ : (κ_{0} ∪κ_{0}^{−})⊕μ_{0}^{−} ≅ (κ_{0} ∪_{θ} κ_{1}^{−}) ∪ (κ_{0} ∪_{θ} κ_{1}^{−})^{−} such that the following diagram commutes.

[figure: κ_{0}; (κ_{0} ∪ κ_{0}^{−}); (κ_{0} ∪ κ_{0}^{−}) ⊕ μ_{0}^{−}; (κ_{0} ∪_{θ} κ_{1}); (κ_{0} ∪_{θ} κ_{1}^{−}) ⊕ (κ_{0} ∪_{θ} κ_{1}^{−})^{−}; j_{0}; i_{0}; Id⊕0; I; Φ; Id⊕0]

Let θ′ : H_{0}^{∗} → H_{1}^{∗} be any homomorphism extending θ. Recall that

(κ_{0} ∪ κ_{0}^{−})(x_{0}, y_{0}) = λ_{0}^{−1}(x_{0}, x_{0} + α_{0}) − λ_{0}^{−1}(y_{0}, y_{0} + α_{0})

on {(x_{0}, y_{0}) ∈ H_{0}^{∗} ⊕ H_{0}^{∗} : π_{0}(x_{0}) = π_{0}(y_{0})}. We define

I((x_{0}, y_{0})) = [(x_{0}, θ′(y_{0})), (y_{0}, θ′(y_{0}))]

and verify that I : κ_{0} ∪ κ_{0}^{−} ↪ (κ_{0} ∪_{θ} κ_{1}^{−}) ⊕ (κ_{0} ∪_{θ} κ_{1}^{−})^{−} is an isometric embedding.

(κ_{0} ∪_{θ} κ_{1}^{−}) ⊕ (κ_{0} ∪_{θ} κ_{1}^{−})^{−}[I(x_{0}, y_{0})]

= λ_{0}^{−1}(x_{0}, x_{0} + α_{0}) − λ_{0}^{−1}(y_{0}, y_{0} + α_{0})

+λ_{1}^{−1}(θ′(y_{0}), θ′(y_{0}) + α_{1}) − λ_{1}^{−1}(θ′(y_{0}), θ′(y_{0}) + α_{1})

= (κ_{0} ∪ κ_{0}^{−})([(x_{0}, y_{0})]

Since I(j_{0}(v)) = [(λ̂_{0}(v), 0), (0, 0)] = (Id ⊕ 0)(i_{0}(v)) the left side of the diagram commutes. Now I(κ_{0} ∪κ_{0}^{−}) is a nonsingular subform of (κ_{0} ∪_{θ} κ_{1}^{−})⊕(κ_{0} ∪_{θ} κ_{1}^{−})^{−} which is also nonsingular and hence there is a nonsingular form μ_{0} and an isometry

Φ : (κ_{0} ∪ κ_{0}^{−}) ⊕ μ_{0}^{−} ≅ (κ_{0} ∪_{θ} κ_{1}^{−}) ⊕ (κ_{0} ∪_{θ} κ_{1}^{−})^{−}

which extends I. Thus the right side of the diagram commutes.

Now as Φ is an isometry Φ(j_{0}(κ_{0})^{⊥}) = (Φ(j_{0}(κ_{0})))^{⊥}. But j_{0}(κ_{0})^{⊥} = j_{1}(κ_{0}^{−}) ⊕ μ_{0}^{−} and (Φ(j_{0}(κ_{0})))^{⊥} = (i_{0}(κ_{0}))^{⊥} = i_{1}(κ_{1}^{−}) ⊕ (κ_{0} ∪_{θ} κ_{1}^{−})^{−}. We therefore define Θ_{1} :=

2. STABLE EQUIVALENCE OF QUADRATIC FUNCTIONS 83

Φ|_{j_{0}(κ_{0})^{⊥}} and then

Θ_{1} : j_{1}(κ_{0}^{−}) ⊕ μ_{0}^{−} ≅ i_{1}(κ_{1}^{−}) ⊕ (κ_{0} ∪_{θ} κ_{1}^{−})^{−}

is an isometry. It follows that Θ := (i_{1}^{−1} ⊕ Id_{κ_{0}∪_{θ}κ_{1}^{−}}) ∘ Θ_{1} ∘ (j_{1} ⊕ Id_{μ_{0}}) is an isometry, Θ : κ_{0} ⊕μ_{0} ≅ κ_{1} ⊕(κ_{0} ∪_{θ} κ_{1}^{−}). It remains to show that Θ induces θ. Let the underlying modules and forms of μ_{0} and κ_{0} ∪_{θ} κ_{1}^{−} be (H_{μ_{0}}, λ_{μ_{0}}) and (H_{κ}, λ_{κ}) respectively. We must determine the right most vertical homomorphism in the following commutative diagram.

[figure: 0; H_{0} ⊕ H_{μ_{0}}; H_{0}^{∗} ⊕ H_{μ_{0}}^{∗}; T G_{0}; 0; 0; H_{1} ⊕ H_{κ}; H_{1}^{∗} ⊕ H_{κ}^{∗}; T G_{1}; 0.; λ̂_{0}⊕λ̂_{μ_{0}}; Θ; π_{0}; (Θ^{∗})^{−1}; ?; λ̂_{1}⊕λ̂_{κ}; π_{1}]

As both μ_{0} and κ_{0} ∪_{θ} κ_{1}^{−} are nonsingular π_{0}|_{H_{μ_{0}}^{∗}} and π_{1}|_{H_{κ}^{∗}} are identically zero and so it suffices to consider Θ|_{H_{0}⊕0} ∘ pr_{H_{1}}. But Θ|_{H_{0}⊕0} = (i_{1}^{−1} ⊕ Id_{κ_{0}∪_{θ}κ_{1}^{−}}) ∘ I ∘ j_{1} and hence Θ|_{H_{0}⊕0} ∘ pr_{H_{1}} = λ̂_{1}^{−1} ∘ θ′ ∘ λ̂_{0}. But now one verifies that (λ̂_{1}^{−1} ∘ θ′ ∘ λ̂_{0})^{∗−1} = θ′. Hence Θ induces the same map on T G_{0} as θ′ which is θ as θ′ was chosen to extend θ. We summarize this argument with the following subdiagram of the previous diagram

[figure: 0; H_{0}; H_{0}^{∗}; T G_{0}; 0; 0; H_{1}; H_{1}^{∗}; T G_{1}; 0.; λ̂_{0}; Θ|_{H_{0}⊕0}; π_{0}; θ′; θ; λ̂_{1}; π_{1}]

Corollary 3.13. Let κ_{0}(H_{0}, λ_{0}, α_{0}) and κ(H_{1}, λ_{1}, α_{1}) be quadratic functions in the family F^{∗}. Suppose that θ is an isometry of the induced quadratic linking families, θ : δ^{∗}(κ_{0}) ≅ δ^{∗}(κ_{1}). Then, there exist nonsingular quadratic functions μ_{0} and μ_{1} and an isometry Θ : κ_{0} ⊕ μ_{0} ≅ κ_{1} ⊕ μ_{1} which induces θ.

Proof. Choose a section Φ_{0} ∈ Sec(τ_{G_{0}}) and let Ψ_{i} ∈ Sec(ρ_{i}), i = 0, 1, be sections such that Φ(Ψ_{0}) = Φ_{0} and Φ(Ψ_{1}) = Φ_{1} := τ_{θ}(Φ_{0}). Then since θ is an isometry of quadratic linking families, θ|_{T G_{0}} is an isometry from q^{∗}(κ_{0})(Φ_{0}) = q^{∗}(κ_{0}(Ψ_{0})) to q^{∗}(κ_{1})(Φ_{1}) = q^{∗}(κ_{1}(Ψ_{1})). Applying Lemma 3.12 to the θ|_{T G_{0}} and setting μ_{1} := (κ_{0}(Ψ_{0}) ∪_{θ|_{T G_{0}}} κ_{1}(Ψ_{1})^{−}) we conclude that there is a nonsingular quadratic function μ_{0} and an isometry

Θ_{N} : κ_{0}(Ψ_{0}) ⊕ μ_{0} → κ_{1}(Ψ_{1}) ⊕ μ_{1}

such that δ^{∗}(Θ_{N}) = θ|_{T G_{0}}. Now κ_{0} = α|_{F_{0}} ⊕ κ(Ψ_{0}) and κ_{1} = α_{1}|_{F_{1}} ⊕ κ(Ψ_{1}) and the isometry Θ_{N} is defined on H(Ψ_{0}) ⊕ H_{μ_{0}} where H_{μ_{0}} is the domain of μ_{0}. Thus it remains to use θ to define an isometry from α_{0}|_{F_{0}} to α_{1}|_{F_{1}}. By the definition of Φ_{i}, π_{i}|_{F_{i}^{∗}(Ψ_{i})} : F_{i}^{∗}(Ψ_{i}) → Φ_{i}(G_{i}^{∗∗}) is an isomorphism for i = 0, 1. Also, by the definition of τ_{θ} and the fact that Φ_{1} = τ_{θ}(Φ_{0}), θ(Φ_{0}(G_{0}^{∗∗})) = Φ_{1}(G_{1}^{∗∗}). Moreover, there are isomorphisms, R_{F_{i}} : F_{i}^{∗}(Ψ_{i}) → F_{i}^{∗}, i = 0, 1 defined by restricting the domain of homomorphisms in F_{i}^{∗}(Ψ_{i})

\begin{aligned}R_{F_{i}} : & F_{i}^{∗}(Ψ_{i}) & → & F_{i}^{∗} \\ & z & ↦ & z|_{F_{i}}. \end{aligned}

We therefore define Θ_{D} : F_{0} → F_{1} so that Θ_{D}^{∗−1} makes the following diagram (in which all morphisms are isomorphisms) commute.

[figure: F_{0}^{∗}; F_{0}^{∗}(Ψ_{0}); Φ_{0}(G_{0}^{∗∗}); F_{1}^{∗}; F_{1}^{∗}(Ψ_{1}); Φ_{1}(G_{1}^{∗∗}); Θ_{D}^{∗−1}; R_{F_{0}}; π_{0}|_{F^{∗}_{0}(Ψ_{0})}; θ|_{Φ_{0}(G{∗∗ \atop 0})}; R_{F_{1}}; π_{1}|_{F_{1}(Ψ_{1})^{∗}}]

2. STABLE EQUIVALENCE OF QUADRATIC FUNCTIONS 85

Under the splittings G_{i} = Φ_{i}(G_{i}^{∗∗}) ⊕ T G_{i} and H_{i}^{∗} = F_{i}^{∗}(Ψ_{i}) ⊕ H_{i}^{∗}(Ψ_{i}) the elements β_{i} and α_{i} are mapped to (Φ_{i}(τ_{G_{i}}(β_{i})), β_{i}−Φ_{i}(τ_{G_{i}}(β_{i}))) and (p_{F_{i}}(Ψ)^{∗}(α|_{F_{i}}), α(Ψ_{i})) respectively (see Sections 2.6 and 2.10). Now π_{i}(α_{i}) = β_{i}, and so π_{i}|_{F_{i}^{∗}(Ψ_{i})}(p_{F_{i}}(Ψ)^{∗}(α|_{F_{i}})) = Φ_{i}(τ_{G_{i}}(β_{i})). Since θ(β_{0}) = β_{1} and θ respects the splittings of G_{0} and G_{1}, θ(Φ_{0}(τ_{G_{0}}(β_{0}))) = Φ_{1}(τ_{G_{1}}(β_{1})). Finally, as R_{F_{i}}(p_{F_{i}}(Ψ)^{∗}(α|_{F_{i}})) = α|_{F_{i}} it follows by chasing the diagram defining Θ_{D} that Θ_{D}^{∗}(α_{1}|_{F_{1}}) = α_{0}|_{F_{0}}. That is, Θ_{D} defines an isometry from α_{0}|_{F_{0}} to α_{1}|_{F_{1}}. By definition δ^{∗}(Θ_{D}) = θ|_{Φ_{0}(G_{0}^{∗∗})}. We therefore define the isometry

Θ := Θ_{D} ⊕ Θ_{N} : κ_{0} = [α_{0}|_{F_{0}} ⊕ κ_{0}(Ψ_{0}) ⊕ μ_{0}] → [α_{1}|_{F_{1}} ⊕ κ_{1}(Ψ_{1}) ⊕ μ_{1}] = κ_{1}

and note that

δ^{∗}(Θ) = δ^{∗}(Θ_{D}) ⊕ δ^{∗}(Θ_{N}) = θ|_{Φ_{0}(G_{0}^{∗∗})} ⊕ θ|_{T G_{0}} = θ.

Having assembled the necessary results we conclude this chapter with the proof of Theorem 3.4 which we restate for the reader’s convenience.

Theorem 3.4. Let κ_{0} and κ_{1} be two quadratic functions in the same family F^{∗}. Then the following are equivalent.

1. κ_{0} is stably F^{∗}-equivalent to κ_{1}, κ_{0} ∼_{F^{∗}} κ_{1}.

2. The quadratic linking families induced by κ_{0} and κ_{1} are isometric, δ^{∗}(κ_{0}) ≅ δ^{∗}(κ_{1}).

3. κ_{0} and κ_{1}^{−} are F^{∗}-orthogonal, κ_{0} ⊥_{F^{∗}} κ_{1}^{−}.

Proof. (1.) ⇒ (2.) The argument should be familiar by now. Let μ_{0} and μ_{1} ∈ F^{∗} be nonsingular and Θ : κ_{0} ⊕ μ_{0} → κ_{1} ⊕ μ_{1} be an isometry. Then by Lemma 2.48, δ^{∗}(Θ) induces an isometry from δ^{∗}(κ_{0} ⊕ μ_{0}) to δ^{∗}(κ_{1} ⊕ μ_{1}). But for i = 0, 1, δ^{∗}(κ_{i} ⊕ μ_{i}) = δ^{∗}(κ_{i}) ⊕ δ^{∗}(μ_{i}) = δ^{∗}(κ_{i}). The last equality holding because μ_{i} induces

2. STABLE EQUIVALENCE OF QUADRATIC FUNCTIONS 86

the trivial quadratic linking family because it is nonsingular. Hence δ^{∗}(Θ) defines an isometry from δ^{∗}(κ_{0}) to δ^{∗}(κ_{1}). (2.) ⇒ (3.) If θ : δ^{∗}(κ_{0}) ≅ δ^{∗}(κ_{1}) is an isometry then by the proof of Corollary 3.13 there are sections Ψ_{0} ∈ Sec(ρ_{0}) and Ψ_{1} ∈ Sec(ρ_{1}) such that θ_{T G_{0}} : κ_{0}(Ψ_{0}) ≅ κ_{1}(Ψ_{1}) and such that there is an isometry Θ_{D} : κ_{0}|_{F_{0}} ≅ κ_{1}|_{F_{1}}. We apply Lemma 3.6 to θ_{T G_{0}} and observe that κ_{0}(Ψ_{0}) and κ_{1}(Ψ_{1})^{−} are orthogonal inside κ_{0}(Ψ_{0}) ∪_{θ|_{T G_{0}}} κ_{1}(Ψ_{1})^{−}. Hence κ_{0} and κ_{1} are orthogonal. (3.) ⇒ (1.) Suppose that κ_{0} ⊥_{F^{∗}} κ_{1}^{−}. This means firstly there is an isometry Θ_{D} : κ_{0}|_{F_{0}} ≅ κ_{1}|_{F_{1}}. Secondly it means that there are sections Ψ_{0} ∈ Sec(ρ_{0}) and Ψ_{1} ∈ Sec(ρ_{1}), a nonsingular quadratic function κ and embeddings i_{0} : κ_{0}(Ψ_{0}) ↪ κ and i_{1} : κ_{1}(Ψ_{1})^{−} ↪ κ such that i_{1}(κ_{1}(Ψ_{1})^{−}) = i_{0}(κ_{0}(Ψ_{0}))^{⊥}. We calculate that

δ^{∗}(κ_{0}(Ψ_{0})) ≅ δ^{∗}(i_{0}(κ_{0}(Ψ_{0})))
≅ δ^{∗}(i_{0}(κ_{0}(Ψ_{0}))^{⊥−}) By Lemma 3.10 3
≅ δ^{∗}(i_{1}(κ_{1}(Ψ_{1})^{−})^{−})
≅ −δ^{∗}(i_{1}(κ_{1}(Ψ_{1})^{−}))
≅ −δ^{∗}(κ(Ψ_{1})^{−})
≅ δ^{∗}(κ(Ψ_{1})).

So by Lemma 3.12 κ_{0}(Ψ_{0}) and κ_{1}(Ψ_{1}) are stably F^{∗}-equivalent. Since κ_{0} = κ_{0}|_{F_{0}} ⊕ κ_{0}(Ψ_{0}), κ_{1} = κ_{1}|_{F_{1}} ⊕ κ_{1}(Ψ_{1}) and κ_{0}|_{F_{0}} ≅ κ_{1}|_{F_{0}} we deduce that κ_{0} and κ_{1} are stably F^{∗}-equivalent.

CHAPTER 4 Classification of the manifolds P

It is an immediate consequence of Corollary 2.26 and Theorem 3.4 that two highly connected manifolds P_{0} and P_{1} are almost diffeomorphic if and only if the quadratic linking families Q(P_{0}) and Q(P_{1}) are isometric. Corollary 2.26 asserts that P_{0} and P_{1} are almost diffeomorphic if and only if the stable quadratic functions [κ(P_{0})] and [κ(P_{1})] are equal and Theorem 3.4 asserts (in part) that [κ(P_{0})] = [κ(P_{1})] if and only if Q(P_{0}) ≅ Q(P_{1}). In Section 2 of this chapter we prove the sharper result that every isometry θ : Q(P_{0}) → Q(P_{1}) is realized as f_{!} for some almost diffeomorphism f : P_{0} → P_{1}. Section 3 is an aside which explains the topology behind the key algebraic Lemmas 3.10 and 3.6. In Section 4 we consider the smooth classification of highly connected (8j − 1)-manifolds.

1. Extensions of diffeomorphisms

In Section 8 we reported the following

Theorem 2.24 (Wilkens’ Thesis 3.2). Let L_{0} and L_{1} be 2k-dimensional handlebodies (k > 2) with boundaries ∂L_{0} = P_{0} and ∂L_{1} = P_{1}. Suppose that f : P_{0} → P_{1} is a diffeomorphism. Then there are handlebodies V_{0}, V_{1} with boundaries the standard sphere and a diffeomorphism g which makes the following diagram commute.

[figure: P_{0}; P_{1}; L_{0}♮V_{0}; L_{1}♮V_{1}; f; g]

1. EXTENSIONS OF DIFFEOMORPHISMS 88

We give Wilkens’ proof of this result both for completeness and because the steps in the proof of Lemma 3.12 are algebraic analogues inspired by the geometric steps in Wilkens’ proof.

Proof. To begin form the manifold M = L_{0} ∪_{f} −L_{1} by identifying points in ∂L_{0} = P_{0} and ∂L_{1} = P_{1} via f. Using Van Kampen’s theorem and the Mayer-Vietoris sequence for M = L_{0} ∪ −L_{1} we see that M is a closed highly connected 2k-manifold. We wish to show that M# − M is the connected sum of the double of L_{0} = L_{0} ∪_{Id} −L_{0} and some other highly connected manifold which will be called M_{0}. To do this we embed a 2^{k}-disc D^{2k} in L_{1} which itself is embedded in M and let N = (M − D^{2k}) × [0, 1]. Then there is a diffeomorphism F_{1} : ∂N ≅ M# − M. Now if we consider the pair (N, L_{0}) where L_{0} is embedded as L_{0} × 0 ↪ (M × 0) then

H_{l}(N, L_{0}) ≅ H_{l}(M − D^{2k}, L_{0}) as N ≃ M − D^{2k}

≅ H_{l}(L_{1} − D^{2k}, P_{1}) by excision

≅ \{ \begin{matrix}H_{k}(L_{1}, P_{1}) & l = k \\ 0 & else \end{matrix} \} .

Thus N has a handlebody decomposition based on L_{0}×[0, 1] with k-handles attached, one for each element of a basis of H_{k}(M−D^{2k}, L_{0}). The attaching maps for the cores of these handles are embeddings S^{k−1} ↪ L_{0}×1 and as L_{0}×1 is (k−1)-connected we may assume after isotopy that the images of the attaching maps all lie in another 2k-disc D_{1}^{2k} ↪ L_{0} × 1. If follows that there is diffeomorphism F_{0} : ∂N ≅ (L_{0} ∪_{Id} −L_{0})#M_{0} where M_{0} is the boundary of the manifold obtained by attaching the j-handles to D_{1}^{2k} ⊂ D^{2k+1}. Notice that since all the necessary isotopies occur in L_{0} × 1 that we have a fixed embedding of L_{0} = (L_{0} × 0) ↪ ∂N. Hence if we define G to be the diffeomorphism G := F_{0}^{−1} ∘ F_{1} we have a commutative diagram of embeddings and

2. ALMOST SMOOTH CLASSIFICATION OF THE MANIFOLDS P 89

diffeomorphisms

[figure: ((L_{0} ∪_{Id} −L_{0}) − D^{2j}); (L_{0} ∪_{Id} −L_{0})#M_{0}; L_{0}; ((L_{0} ∪_{f} −L_{1}) − D^{2j}); (L_{0} ∪_{f} −L_{1})# − M; G; i_{0}; i_{1}]

such that G|_{i_{0}(L_{0})} = Id. If follows that G defines a diffeomorphism

G|_{−L_{0}#M_{0}} : −L_{0}#M_{0} ≅ −L_{1}# − M

of the complements of i_{0}(L_{0}) and i_{1}(L_{0}). Therefore G|_{−L_{0}#M_{0}} induces a diffeomorphism on the boundary which must be f since we have the following commutative square.

[figure: ∂L_{0}; −∂L_{0}; ∂L_{0}; −∂L_{1}; Id; Id; G|_{−∂L_{0}}; f]

Remark 4.1. Wilkens’ theorem is, in particular, a result about bordism of diffeomorphisms which is a topic that has since received significant attention. In particular see Kreck [Kr1] and Quinn [Q]. One might hope to prove this theorem from a more general point of view by arguing first that the bordism of diffeomorphisms over P are trivial and then performing surgery on the cobounding diffeomorphism to make it a diffeomorphism of highly connected manifolds.

2. Almost smooth classification of the manifolds P

The following theorem combines the statements of Theorem B and 1.5.

2. ALMOST SMOOTH CLASSIFICATION OF THE MANIFOLDS P 90

Theorem 4.2. For every finitely generated abelian group G and every characteristic (resp. even) quadratic linking family Q defined on G, there is a highly connected (8j−1)-manifold P, j = 1, 2 (resp j > 2), such that Q(P) ≅ Q. The quadratic linking family Q(P) of a highly connected manifold P is a complete invariant of almost diffeomorphisms. That is, if P_{0} and P_{1} are two highly connected (8j − 1)-manifolds then there is an almost diffeomorphism f : P_{0} → P_{1} such that f_{!} = θ : H^{4j}(P_{0}) ≅ H^{4j}(P_{1}) if and only if θ defines an isometry of quadratic linking families from Q(P_{0}) to Q(P_{1}).

Proof. Let Q = (G, q^{∗}, β) be a quadratic linking family and let Φ be any section in Sec(τ_{G}). Then G = Φ(G^{∗∗}) ⊕ T G where T G is the torsion subgroup of G. In the characteristic case Corollary 22.36 ensures that there is a nondegenerate quadratic function κ′ = κ(H′, λ′, α′) ∈ F^{c} and an isomorphism θ′ : Cok(λ̂′) → T G such that q^{c}(κ′) = q^{c}(Φ) ∘ θ. We define κ := κ′ ⊕ τ_{G}(β) : H′ ⊕ G^{∗} → ℤ and leave the reader to verify that θ ⊕ Φ : Cok(λ̂) ⊕ G^{∗∗} → G defines an isometry from δ^{c}(κ) to Q. The argument in the even case is similar but applies instead Theorem 2.35 to deduce the existence of an nondegenerate even quadratic function such κ such that q^{ev}(κ) ≅ q^{ev}(Φ).

Now let L_{i} be handlebodies such that ∂L_{i} = P_{i}#Σ_{P_{i}}. Suppose that there is an isometry

θ : Q(P_{0}) = δ^{∗}(κ(L_{0})) → δ^{∗}(κ(L_{1})) = Q(P_{1}).

By Corollary 3.13 there is a nonsingular quadratic function μ_{0} and an isometry

Θ : κ(L_{0}) ⊕ μ_{0} → κ(L_{1}) ⊕ κ_{0} ∪_{θ} κ_{1}^{−}

which induces θ, δ^{∗}(Θ) = θ. Since μ_{0} is nonsingular the boundary of L(μ_{0}) is a homotopy sphere which we shall denote by Σ_{0}. Now, by Wall’s classification of handlebodies Θ is realized by a diffeomorphism

g(Θ) : L_{0}♮L(μ_{0}) → L_{1}♮L(κ_{0} ∪_{θ} κ_{1}^{−}).

3. THE MANIFOLDS L(κ_{0} ∪_{θ} κ_{1}^{−}) AND L_{0} ∪_{f^{o}} −L_{1} 91

Restricting g to the boundary of L_{0}♮L(μ_{0}) we obtain a diffeomorphism

f : P_{0}#Σ_{P_{0}}#Σ_{0} → P_{1}#Σ_{P_{1}}

which defines an almost diffeomorphism f′ : P_{0} → P_{1} such that f′_{!} = f_{!} = θ.

Conversely, suppose that f : P_{0} → P_{1} be an almost diffeomorphism. By Proposition 2.1 there is a homotopy sphere Σ and a diffeomorphism f′ : P_{0}#Σ → P_{1} which agrees with f outside a disc. Then Σ_{P_{1}} = Σ_{P_{0}#Σ} = Σ_{P_{0}}#Σ_{Σ} and there is a diffeomorphism

f′′ : P_{0}#Σ#Σ_{P_{1}} = P_{0}#Σ_{P_{0}}#Σ#Σ_{Σ} → P_{1}#Σ_{P_{1}}.

Let L_{Σ} be a handlebody bounding Σ#Σ_{Σ}. Applying Theorem 2.24 to f′′ : ∂(L_{0}♮L_{Σ}) → ∂L_{1} we deduce that there are handlebodies V_{0} and V_{1} cobounding standard spheres and a diffeomorphism

h : L_{0}♮L_{Σ}♮V_{0} → L_{1}♮V_{1}

which extends f′′. The map induced by h, Θ(h) : κ(L_{0}) ⊕ κ(L_{Σ}♮V_{0}) → κ(L_{1}) ⊕ κ(V_{1}), is an isometry. By Lemma 2.48 it follows that

f!^{′′} : δ^{∗}(κ(L_{0})) ⊕ δ^{∗}(κ(L_{Σ}♮V_{0})) → δ^{∗}(κ(L_{1})) ⊕ δ^{∗}(κ(V_{1}))

is an isometry. However as V_{0} and L_{Σ}♮V_{0} are both bounded by homotopy spheres, δ^{∗}(κ(V_{0})) and δ^{∗}(κ(L_{Σ}♮V_{1})) are both the trivial quadratic linking family and therefore f!^{′′} defines an isometry from δ^{∗}(κ(L_{0})) = Q(P_{0}) to δ^{∗}(κ(L_{1})) = Q(P_{1}). Finally, f!^{′′} = f_{!} since we may assume that the two maps differ only on the interior of a disc.

3. The manifolds L(κ_{0} ∪_{θ} κ_{1}^{−}) and L_{0} ∪_{f^{o}} −L_{1}

Let P_{0} and P_{1} be the boundaries of handlebodies and let f : P_{0} → P_{1} be an almost diffeomorphism. Recall that by Proposition 2.1 that there is a homotopy sphere Σ, a disc D^{8j−1} ↪ P_{0} and a diffeomorphism f′ : P_{0}#Σ → P_{1} such that f′|_{P_{0}−D^{8j−1}} = f|_{P_{0}−D^{8j−1}} (where the connected sum of Σ and P_{0} is understood to

3. THE MANIFOLDS L(κ_{0} ∪_{θ} κ_{1}^{−}) AND L_{0} ∪_{f^{o}} −L_{1} 92

occur inside D^{8j−1}). We define f^{o} to be the restriction f^{o} := f|_{P_{0}−D^{8j−1}} which is a diffeomorphism onto its image.

Lemma 4.3. Let L_{0} = L(κ_{0}) and L_{1} = L(κ_{1}) be handlebodies with associated nondegenerate quadratic functions κ_{0} = κ(H_{0}, λ_{0}) and κ_{1} = κ(H_{1}, λ_{1}). Let P_{0} and P_{1} be the boundaries of L_{0} and L_{1}, let f : ∂P_{0} → ∂P_{1} be an almost diffeomorphism which induces f_{!} = θ : H^{4j}(P_{0}) → H^{4j}(P_{1}) and let Σ and f^{o} be as above. Then θ : Q(P_{0}) ≅ Q(P_{1}) is an isometry of quadratic linking families and L_{0} ∪_{f^{o}} −L_{1} is diffeomorphic to κ_{0} ∪_{θ} κ_{1}^{−}.

Proof. The map f^{o} is a diffeomorphism onto its image and, after perhaps rounding corners, the adjunction space L := L_{0}∪_{f^{o}}−L_{1} is a smooth manifold with boundary. The boundary of L is obtained by gluing the disc D^{8j−1} ⊂ P_{0} to a disc D_{1}^{8j−1} which is the complement of the range of f^{0}. Thus ∂L is a homotopy sphere. Now let P^{o} be the subset P_{0} −D^{8j−1} = f^{o}(P_{0} −D^{8j−1}) ⊂ P_{1} contained in L := L_{0} ∪_{f^{o}} −L_{1}. Applying Van-Kampen’s theorem and the Mayer-Vietoris sequence to the decomposition of L into the triple (L_{0}, L_{1}, P^{o}) we conclude that L is (4j − 1) connected. It follows that L falls under Wall’s classification of handlebodies and we let κ = κ(H, λ, α) be the quadratic function of L. We will show that κ is isometric to κ_{0} ∪_{θ} κ_{1}^{−}. There are embeddings of i_{0} : H_{0} ↪ H and i_{1} : H_{1} ↪ H embedding κ_{0} and κ_{1}^{−} respectively into κ corresponding to the obvious embeddings of L_{0} and −L_{1} into L_{0} ∪_{f^{o}} −L_{1}. We claim that that κ_{0}^{⊥} = κ_{1}^{−} because firstly it is clear that spheres in L_{0} representing homology classes do not intersect spheres in L_{1} and vice versa. Secondly, suppose that v̄ : S^{4j} ↪ L represents v ∈ H_{4j}(L) and that λ(v, v_{0}) = 0 for every v_{0} ∈ i_{0}(H_{0}). It follows via the Whitney trick that v̄(S^{4j}) may be isotoped away from all the handles of L_{0} and thence into −L_{1}. Therefore v ∈ i_{1}(H_{1}). Applying Lemma 3.10 we conclude

3. THE MANIFOLDS L(κ_{0} ∪_{θ} κ_{1}^{−}) AND L_{0} ∪_{f^{o}} −L_{1} 93

that κ ≅ κ_{0} ∪_{θ_{κ}} κ_{1}^{−} and that

θ_{κ} : Q(P_{0}) = δ^{∗}(κ_{0}) → δ^{∗}(κ_{1}) = Q(P_{1})

is an isometry of quadratic linking families. It remains therefore to demonstrate that θ_{κ} = θ.

From the proof of Lemma 3.10 recall the projections π_{i} : H_{i}^{∗} → Cok(λ̂_{0})T G_{0} = H^{4j}(P_{0}), i = 0, 1 and recall that θ_{κ} : T G_{0} → T G_{1} is the unique isomorphism such that (π_{0} ⊕ π_{1})(i_{0}^{∗} ⊕ i_{1}^{∗})(H^{∗}) ⊂ T G_{0} ⊕ T G_{1} is the graph of θ_{κ}. Now let j_{0} : P^{o} ↪ L_{0} and j_{1} : P^{o} ↪ −L_{1} denote the inclusions. The usual Mayer-Vietoris arguments show that there is a long exact sequence

0 ⟶ H^{4j}(L) ⟶^{i_{0}^{∗}⊕i_{1}^{∗}} H^{4j}(L_{0}) ⊕ H^{4j}(−L_{1}) ⟶^{j_{0}^{∗}−j_{1}^{∗}} H_{4j−1}(P^{o}) ⟶ 0.

Identifying H^{4j}(P^{o}) = H^{4j}(P_{0} − D^{8j−1}) we see that (j_{0}^{∗} − j_{1}^{∗})(x_{0}, x_{1}) = [x_{0}] − f^{∗}[x_{1}]. Hence

\begin{aligned}(π_{0} ⊕ π_{1})(i_{0}^{∗} ⊕ i_{1}^{∗})(H^{∗}) & = \{([x_{0}], [x_{1}])|f^{∗−1}([x_{0}]) = [x_{1}]\} \\ & = \{([x_{0}], [x_{1}])|θ([x_{0}]) = [x_{1}]\} \end{aligned}

and θ = θ_{κ} as required.

Given the manifold L(κ_{0} ∪_{θ} κ_{1}^{−}) with boundary Σ we let D^{8j} denote the almost smooth disc obtained by taking the cone of Σ. Hence D^{8j} is a P L manifold equipped with a smooth structure except at the cone point which we denote by p_{0}. Now form M(κ_{0} ∪_{θ} κ_{1}^{−}) := L(κ_{0} ∪_{θ} κ_{1}^{−}) ∪_{Σ} D^{8j} which is an almost smooth manifold with singular point p_{0}.

Lemma 4.4. Let L_{0} = L(κ_{0}) and L_{1} = L(κ_{1}) be handlebodies associated to nondegenerate quadratic functions κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}). If θ : δ^{∗}(κ_{0}) ≅ δ^{∗}(κ_{1}) is an isomorphism of the induced quadratic linking families then

1. there is an embedding of the disjoint union L(κ_{0}) ⊔ −L(κ_{1}) ↪ M(κ_{0} ∪_{θ} κ_{1}^{−}) with complement an almost smooth h-cobordism from ∂L_{0} to −∂L_{1},

3. THE MANIFOLDS L(κ_{0} ∪_{θ} κ_{1}^{−}) AND L_{0} ∪_{f^{o}} −L_{1} 94

2. there is an almost diffeomorphism f(θ) : ∂L_{0} ≅ ∂L_{1} realizing θ on homology,

3. the manifold M(κ_{0} ∪_{θ} κ_{1}^{−}) is almost diffeomorphic to the manifold L_{0} ∪_{f(θ)} −L_{1} formed by gluing L_{0} to −L_{1} with f(θ).

Proof. 1. Recall that the group H = {(x_{0}, x_{1}) ∈ H_{0}^{∗} ⊕ H_{1}^{∗}|θ([x_{0}]) = [x_{1}]} is the domain κ_{0} ∪_{θ} κ_{1}^{−} and that there are isometric embeddings

\begin{aligned}i_{0} : & H_{0} & ↪ & H \\ & v_{0} & ↦ & (λ̂_{0}(v_{0}), 0) \end{aligned}  \begin{aligned}i_{1} : & H_{1} & ↪ & H \\ & v_{1} & ↦ & (0, −λ̂_{1}(v_{1})). \end{aligned}

from Lemma 3.6. For j = 0, 1, the group i_{j}(H_{j}) is a summand of H for it is the kernel of the homomorphism H → H_{j}^{∗} which sends a pair (x_{0}, x_{1}) ∈ H to x_{j} ∈ H_{j}^{∗}. Thus if {v_{01}, …, v_{0m}} and {v_{11}, …, v_{1n}} are bases for H_{0} and H_{1} respectively, then there are bases {y_{1}, …, y_{N}} and {z_{1}, …, z_{N}} for H such that i_{0}(v_{0j}) = y_{j} for 1 ≤ j ≤ l and i_{1}(v_{1j}) = z_{j} for 1 ≤ j ≤ m. According to Wall’s Theorem 2.4 the handlebody L := L(κ_{0} ∪_{θ} κ_{1}^{−}) has a presentation for each of the bases {y_{i}} and {z_{i}}. In fact, Wall proved the following. Let D^{8j} ↪ L be a fixed disc in the interior of L and let K be the closure of L − D^{8j}. Then

H_{4j}(L) ≅ H_{4j}(L, D^{8j}) ≅ H_{4j}(K, ∂D^{8j}) ≅ π_{4j}(K, ∂D^{8j}).

If {z′_{1}, …, z′_{n}} is any base for H_{4j}(L) then for each z′_{i} there are pairwise disjoint embeddings of h̄_{i} : (D^{4j}, ∂D^{4j}) ↪ (K, ∂D^{8j}) which are transverse to ∂D^{8j} along h̄_{i}(∂D^{4j}) and which represent the element in π_{4j}(K, ∂D^{8j}) corresponding to z′_{i}. These embeddings can be thickened to handles h_{i} : (D^{4j} × D^{4j}, ∂D^{4j} × D^{4j}) ↪ (K, ∂D^{8j}) and the resulting handle body L′ ⊂ L obtained from the union of D^{8j} and the union of the images of the h_{i} is diffeomorphic to L. In fact L − L′ is an h-cobordism from ∂L to ∂L′. We now choose disjoint discs D_{0}^{8j} ↪ L and D_{1}^{8j} ↪ L one for each basis {y_{i}} and {z_{i}} and attach handles h_{0i} and h_{1j} to these discs as above but only for basis elements λ̂_{0}(v_{0j}) and λ̂_{1}(v_{1j}). The manifolds D_{0}^{8j} ∪ (∪_{i=1}^{l}h_{0i}) and D^{8j} ∪ (∪_{j=1}^{m}h_{1j}) thus

3. THE MANIFOLDS L(κ_{0} ∪_{θ} κ_{1}^{−}) AND L_{0} ∪_{f^{o}} −L_{1} 95

define embeddings of L_{0} and −L_{1} into L. Recall that λ denotes the intersection form of L and that

λ[(λ̂_{0}(v_{0}), 0), (0, −λ̂_{1}(v_{1})] = 0

for every v_{0} ∈ H_{0} and v_{1} ∈ H_{1}. It follows that the handles h_{0i} and h_{1j} have algebraic intersection zero and so by the Whitney trick they may be isotoped to have empty geometric intersection. We may therefore assume that L(κ_{0}) and −L(κ_{1}) are disjointly embedded into L(κ_{0} ∪_{θ} κ_{1}^{−}) ⊂ M(κ_{0} ∪_{θ} κ_{1}^{−}) and we use the same symbols to denote the disjointly embedded manifolds.

Let N := M(κ_{0} ∪_{θ} κ_{1}^{−}) − Int(L_{0} ⊔ −L_{1}) be the complement of the interior of L_{0} ⊔ −L_{1} inside M(κ_{0} ∪_{θ} κ_{1}^{−}) which is a codimension zero submanifold of M. We orient N compatibly with M and then ∂N = ∂L_{0} ⊔ −∂L_{1}. Let J : L_{0} ⊔ −L_{1} ↪ M, i_{N} : N ↪ M and i_{∂N} : ∂N ↪ L_{0} ⊔ −L_{1} denote the obvious inclusions and E^{∗} : H^{∗}(M, L_{0} ⊔ −L_{1}) = H^{∗}(N, ∂N) the excision isomorphisms. The long exact cohomology sequences of the pairs (N, ∂N) and (M, L_{0} ⊔ −L_{1}) are linked in the following commutative diagram (where we abbreviate L_{0} ⊔ −L_{1} to L_{0⊔1}).

[figure: H^{0}(N, ∂N); H^{0}(N); H^{0}(∂N); H^{1}(N, ∂N); H^{1}(N); H^{0}(M, L_{0⊔1}); H^{0}(M); H^{0}(L_{0⊔1}); H^{1}(M, L_{0⊔1}); H^{1}(M); 0; ℤ; ℤ ⊕ ℤ; ℤ; 0; E^{0}; J^{0}; i_{N}^{0}; i_{∂N}^{0}; E^{1}; i_{N}^{0}; =; =; =; =; =]

As E^{0}, I_{∂N}^{0} and E^{1} are all isomorphisms we conclude that i_{N}^{0} is an isomorphism and hence that N is connected. A simple application of Van Kampen’s theorem shows that N is also simply connected. Applying Poincaré duality to the manifold N we have that H_{8j−∗}(N) ≅ H^{∗}(N, ∂N) ≅ H^{∗}(M, L_{0} ⊔ −L_{1}). We may calculate the groups H^{∗}(M, L_{0} ⊔ −L_{1}) again using the long exact cohomology sequence of the pair

3. THE MANIFOLDS L(κ_{0} ∪_{θ} κ_{1}^{−}) AND L_{0} ∪_{f^{o}} −L_{1} 96

(M, L_{0} ⊔ −L_{1}). This sequence is zero below dimension (8j − 1) except in dimensions 4j and 4j + 1. The following diagram shows the relative cohomology sequences for the pairs (M, L_{0} ⊔−L_{1}) and (N, ∂N) on the left. On the right it shows an isomorphic sequence extending the right hand side of Diagram 3.7 in Lemma 3.6 from where we recall the projections π_{0} : H_{0}^{∗} → T G_{0}, π_{1} : H_{1}^{∗} → T G_{1}, π : H → T G = {([x_{0}], θ[x_{0}]} ⊂ T G_{0} ⊕ T G_{1} and Π : H_{0}^{∗} ⊕ H_{1}^{∗} → T G^{∧}.

Diagram 4.5.

[figure: 0; 0; 0; 0; H^{4j}(M); H^{4j}(N); H; T G; H^{4j}(L_{0} ⊔ −L_{1}); H^{4j}(∂N); H_{0}^{∗} ⊕ H_{1}^{∗}; T G_{0} ⊕ T G_{1}; H^{4j+1}(M, L_{0} ⊔ −L_{1}); H^{4j+1}(N, ∂N); T G^{∧}; T G^{∧}; 0; 0; 0; 0; i_{N}^{∗}; J^{∗}; I^{∗}; π; i; δ_{1}; J_{0}^{∗}⊕J_{1}^{∗}; δ_{2}; π_{0}⊕π_{1}; Π; E^{4j+1}]

Applying Poincaré duality and the universal coefficient theorem for torsion groups we conclude that

H^{∗}(N) ≅ \{ \begin{matrix}ℤ & ∗ = 0, 8j − 1 \\ T G & ∗ = 4j \\ 0 & else. \end{matrix} \}

3. THE MANIFOLDS L(κ_{0} ∪_{θ} κ_{1}^{−}) AND L_{0} ∪_{f^{o}} −L_{1} 97

With the identifications H^{4j}(∂L_{0}) = T G_{0}, H^{4j}(−∂L_{1}) = T G_{1}, H^{4j}(N) = T G and H^{4j}(∂N)T G_{0} ⊕ T G_{1} and the notations I_{0} : ∂L_{0} ↪ N and I_{1} : −∂L_{1} ↪ N for the standard inclusions we conclude also that for x = ([x_{0}], θ[x_{0}]) ∈ H^{4j}(N), I_{0}^{∗}(x) = [x_{0}] and I_{1}^{∗}(x) = θ[x_{0}]. Given our calculation of H^{∗}(N) it follows that both I_{0} and I_{1} are I_{0} and I_{1} are (co)homology equivalences of simply connected manifolds. So by Whitehead’s theorem both I_{0} and I_{1} are homotopy equivalences. Thus N is an hcobordism from ∂L_{0} to −∂L_{1}. It is an almost smooth manifold since the singular point of M in contained in N. (2.) The almost smooth manifold N has one singular point with associated exotic sphere Σ := ∂L(κ_{0} ∪_{θ} κ_{1}^{−}). As described in Section 1 we may construct a smooth h-cobordism from ∂L_{0}#Σ to ∂L_{1} by thickening a path from a point on Σ to ∂L_{0} and then cutting out this thickened path to produce N′ ⊂ N. We note that cutting out this path has no effect on the 4jth cohomology groups of the manifolds involved. We denote the inclusions of the boundary components of the modified manifold by I′_{0} : ∂L_{0}#Σ ↪ N′ and I′_{1} : −∂L_{1} ↪ N′. Now let F : N′ → (∂L_{0}#Σ) × [0, 1] be a diffeomorphism which restricts to the identity on ∂L_{0}#Σ. The diffeomorphism at the other end of N′ is a diffeomorphism F_{1} := F ∘ I′_{1} : −∂L_{1} → −(L_{0}#Σ). Given x ∈ H^{4j}(N′) we have from above that I_{1}^{′∗}(x) = θI_{0}^{′∗}(x). But since I′_{0} ∘ F |_{∂L_{0}#Σ} = Id_{∂L_{0}#Σ} we calculate that

F_{1}^{∗}(I_{0}^{′∗}(x)) = (F ∘ I′_{1})^{∗}((I′_{0})^{∗}(x))
= I_{1}^{′∗}(F^{∗}(I_{0}^{′∗}(x)))
= I_{1}^{′∗}((I′_{0} ∘ F_{∂L_{0}#Σ})^{∗}(x))
= I_{1}^{′∗}(x)
= θ(I_{0}^{∗′}(x)).

Thus F_{1} induces θ on cohomology. Therefore we take f(θ) to be the map F_{1}^{−1} : ∂L_{0}#Σ → ∂L_{1} which we may consider as an almost diffeomorphism from ∂L_{0} to

4. SMOOTH CLASSIFICATION OF THE MANIFOLDS P 98

∂L_{1} with one singular point whose associated exotic sphere is Σ. The isomorphism f(θ)_{!} = (F_{1}^{−1})^{∗−1} = F_{1}^{∗} = θ. (3.) The almost diffeomorphism f(θ) has one singular point so let f^{o} = f(θ)|_{∂L_{0}−D^{8j−1}}. By Lemma 4.3 the adjunction space L_{0} ∪_{f^{o}} −L_{1} space is diffeomorphic to L(κ_{0} ∪_{θ} κ_{1}^{−}). If we now form the adjunction space L_{0} ∪_{f(θ)} −L_{1} then the effect is to identify the two discs from which the homotopy sphere Σ = ∂(L_{0} ∪_{f^{o}} −L_{1}) is formed. But this has the same effect as coning off Σ to produce M(κ_{0} ∪_{θ} κ_{1}^{−}). Hence M(κ_{0} ∪_{θ} κ_{1}^{−}) and L_{0} ∪_{f(θ)} −L_{1} are diffeomorphic except at one singular point.

Remark 4.6. Note that for highly connected rational homology spheres bounding handlebodies Lemmas 4.3 and 4.4 together give a proof that the quadratic linking function is a well defined and complete invariant of almost diffeomorphisms. This second proof is independent of Theorem 2.24. Its engine is the h-cobordism theorem which is the heart of Lemma 4.3. We note also that with a little care this alternate proof could be extended to all highly connected (8j − 1)-manifolds and thus give a second proof of Theorem 4.2 which is independent of Theorem 2.24.

4. Smooth classification of the manifolds P

Recall Corollary 2.26 which states that a highly connected (8j − 1)-manifold P is classified up to almost diffeomorphism by its stable quadratic function [κ(P)]. We begin this section by defining a smooth stable quadratic function for P and showing that it is a complete diffeomorphism invariant of those P which bound handlebodies. We shall call a quadratic function μ ∈ F^{8j} spherical if and only if the boundary of L^{8j}(μ) is diffeomorphic to the standard sphere. Now we have already noted in the proof of Lemma 2.16 that every homotopy sphere Σ bounding a handlebody lies in bP_{8j} and by Theorem 2.55, Σ is standard if and only if s_{1}(Σ) = 0 ∈ ℚ/ℤ. We

4. SMOOTH CLASSIFICATION OF THE MANIFOLDS P 99

can therefore characterize spherical quadratic functions completely algebraically as follows.

Lemma 4.7. A quadratic function μ(H, λ, α) ∈ F^{8j} is spherical if and only if it is nonsingular and

s_{1}(∂L(μ)) = (1/|bP_{8j}|) (S̃_{(j,2j)}λ^{−1}(α, α) − \frac{1}{8}σ(μ)) = 0 ∈ ℚ/ℤ.

We shall call two quadratic functions κ_{0} and κ_{1} in F^{8j} smoothly equivalent if there are spherical quadratic functions μ_{0} and μ_{1} in F^{8j} and an isometry Θ : κ_{0} ⊕μ_{0} ≅ κ_{1} ⊕μ_{1}. It is not difficult to see that smooth equivalence is in fact an equivalence relation. So given a highly connected (8j −1)-manifold we define the smooth quadratic function of P, [κ(P)]_{F^{8j}}^{s}, to be the smooth equivalence class of κ(L) where L is any handlebody such that ∂L = P #Σ_{P}.

Theorem 4.8. Two highly connected (8j −1)-manifolds P_{0} and P_{1} are diffeomorphic if and only if

Σ_{P_{0}} ≅ Σ_{P_{1}} and [κ(P_{0})]_{F^{8j}}^{s} = [κ(P_{1})]_{F^{8j}}^{s}.

Proof. That the manifolds P_{0}#Σ_{P_{0}} and P_{1}#Σ_{P_{1}} are diffeomorphic if and only if [κ(P_{0})]_{F^{8j}}^{s} = [κ(P_{1})]_{F^{8j}}^{s} follows from Wilkens’ Theorem 2.28 and Wall’s classification of handlebodies in a manner completely analogous to Corollary 2.26. The theorem now follows immediately. The determination of smooth equivalence for nondegenerate quadratic functions is a straightforward matter and this allows us to give a more explicit diffeomorphism classification for rational homology spheres.

Theorem 4.9. Highly connected (8j − 1)-dimensional rational homology spheres P_{0} and P_{1} are diffeomorphic if and only if

Q(P_{0}) ≅ Q(P_{1}), s_{1}(P_{0}) = s_{1}(P_{1}) and Σ_{P_{0}} ≅ Σ_{P_{1}}.

4. SMOOTH CLASSIFICATION OF THE MANIFOLDS P 100

Moreover, if P_{0} and P_{1} are diffeomorphic then every isometry θ : Q(P_{0}) → Q(P_{1}) is realized by a diffeomorphism.

Proof. By Theorem B, P_{0} and P_{1} are almost diffeomorphic if and only if Q(P_{0}) ≅ Q(P_{1}). So we may suppose that there is a homotopy sphere Σ such that P_{0} is diffeomorphic to P_{1}#Σ. But then

s_{1}(Σ) = s_{1}(P_{0}) − s_{1}(P_{1}) and Σ_{Σ} ≅ Σ_{P_{0}} − Σ_{P_{1}}

and the first assertion follows since Σ is diffeomorphic to the standard sphere if and only if s_{1}(Σ) = 0 and Σ_{Σ} ≅ S^{8j−1}.

Now suppose that P_{0} and P_{1} are diffeomorphic and that θ : Q(P_{0}) → Q(P_{1}) is an isometry. By Lemma 3.12 there is a nonsingular quadratic function μ_{0} ∈ F^{8j} and an isometry

Θ : κ_{0} ⊕ μ_{0} → κ_{1} ⊕ κ_{0} ∪_{θ} κ_{1}^{−}

where κ_{i} = κ(L_{i}) and L_{i} is a handlebodies with ∂L_{i} = P_{i}#Σ_{P_{i}}, i = 0, 1. Of course by Wall’s classification of handlebodies, Θ is realized by a diffeomorphism g : L_{0}♮L(μ_{0}) ≅ L_{1}♮L(κ_{0} ∪_{θ} κ_{1}^{−}) and so it suffices to show that μ_{0} and κ_{0} ∪_{θ} κ_{1}^{−} are spherical. By Lemma 4.7 this means showing that s_{1}(∂L(μ_{0})) = 0 = s_{1}(∂L(κ_{0} ∪_{θ} κ_{1}^{−})). But now if κ_{0} ∪_{θ} κ_{1}^{−} = κ(H, λ, α) and κ_{i} = κ(H_{i}, λ_{i}, α_{i}) then

σ(κ_{0} ∪_{θ} κ_{1}^{−}) = σ(κ_{0}) − σ(κ_{1}) and λ^{−1}(α, α) = λ_{0}^{−1}(α_{0}, α_{0}) − λ_{1}^{−1}(α_{1}, α_{1}).

It follows immediately that

s_{1}(∂L(κ_{0} ∪_{θ} κ_{1}^{−})) = s_{1}(∂L_{0}) − s_{1}(∂L_{1}) = s_{1}(P_{0}) − s_{1}(P_{1}) = 0

and then we have

s_{1}(∂L(μ_{0})) = s_{1}(∂L_{1}) + s_{1}(∂(L(κ_{0} ∪_{θ} κ_{1}^{−})) − s_{1}(∂L_{0}) = 0.

4. SMOOTH CLASSIFICATION OF THE MANIFOLDS P 101

The question of the smooth equivalence of degenerate quadratic functions is a difficult algebraic problem. It is equivalent to calculating the the inertia group of the boundaries of the corresponding handlebodies. The inertia group of a manifold M, I(M) is the subgroup of the group of homotopy spheres for which Σ#M is diffeomorphic to M. If P is a highly connected (8j − 1)-manifold then I(P) ⊂ bP_{8j} for if [Σ] ≠ 0 ∈ Ω_{8j−1}^{HC} then [P] ≠ [P #Σ] and so P and P #Σ cannot be diffeomorphic. If in addition P is a rational homology sphere then I(P) = 0 for in this case s_{1} is defined for P and detects the addition of any exotic sphere in bP_{8j}. In cite [Wi1] and [Wi3] Wilkens investigated the inertia groups of highly connected manifolds of dimension 7 and 15 and discovered them to be quite subtle. In particular, he found examples where I(P_{0}) = 0 = I(P_{1}) but I(P_{0}#P_{1}) ≠ 0. Wilkens concluded [Wi3] with a conjecture concerning the inertia groups of highly connected manifolds of dimension 7 and 15 which we have been unable to prove or disprove.

CHAPTER 5 Linking forms and linking functions

We have now classified highly connected manifolds in dimension 7 and 15 up to almost diffeomorphism by their quadratic linking functions which need not be homogeneous. Thus the existence of bundles of Hopf invariant one leads us to a somewhat more general class of quadratic object in order to classify manifolds. In this chapter we review the classifications of symmetric bilinear forms on finite abelian groups started by Wall and finished by Kawauchi and Kojima. We next recall Nikulin’s classification of quadratic linking forms refining a given bilinear form and then generalize Nikulin’s result to all quadratic linking functions. Throughout this chapter G shall denote a finite abelian group.

1. Bilinear forms on finite abelian groups

Recall that a nondegenerate symmetric bilinear form b on a finite abelian group G is called a linking form (G, b). Two linking forms (G_{0}, b_{0}) and (G_{1}, b_{1}) are isometric if there is an isomorphism of abelian groups θ : G_{0} ≅ G_{1} such that b_{0} = b_{1} ∘ (θ × θ). The block sum of two linking forms (G_{1}, b_{1}) and (G_{2}, b_{2}) is defined in the obvious way to be (G_{1} ⊕ G_{2}, b_{1} ⊕ b_{2}). If H is a sub-group of G we let H^{⊥} denote the annihilator of H,

H^{⊥} = {g ∈ G|b(g, h) = 0 ∀h ∈ H}.

In Section 7 we promised a proof of the following fundamental lemma which (according to Wall [Wa3]) is well know and may be found in [Wa3].

1. BILINEAR FORMS ON FINITE ABELIAN GROUPS 103

Lemma 5.1. Let (G, b) be a linking form and H a subgroup of G such that b|_{H×H} : H × H → ℚ/ℤ is nonsingular. Then (H^{⊥}, b|_{H^{⊥}×H^{⊥}}) is nonsingular and (G, b) splits as the block sum (H, b|_{H×H}) ⊕ (H^{⊥}, b|_{H^{⊥}×H^{⊥}}).

Proof. Let R_{H} : G^{∧} → H^{∧} be the map which restricts the domain of an element in the torsion dual of G to the subgroup H. Now consider the composition of the adjoint map of b with R_{H},

R_{H} ∘ b̂: G → H^{∧}.

By definition H^{⊥} is the kernel of R_{H}∘b̂ and by hypothesis (R_{H}∘b̂)|_{H} is an isomorphism. Thus H ∩ H^{⊥} = {e} and H and H^{⊥} generate G which is therefore the direct sum of H and H^{⊥}. As b(h, h′) = 0 for all h ∈ H and h′ ∈ H^{⊥} the form b splits over H ⊕ H^{⊥} and it follows that b|_{H^{⊥}} must be nonsingular for if it were singular then b would also be singular.

Let p be a prime. Every linking form splits as a block sum over the p-primary components of its domain. We therefore let N_{p} denote the semi-group of linking forms on p-groups under block sum and N denote abelian semi-group of all linking forms under block sum. Hence N ≅ ⨁_{p}N_{p}. The structure of linking forms for p an odd prime is relatively simple. In this case every linking form splits as the block sum of linking forms over cyclic groups ℤ/p^{k}. The following notation comes from Wall [Wa3] and Kawauchi and Kojima [KK]. The form, A_{p}^{k}(θ) denotes the unique (up to isometry) linking form (ℤ/p^{k}, b) with b(e, e) = θ where e is a generator of ℤ/p^{k}. When p is understood we write simply A^{k}(θ). If p is an odd prime there are two isometry classes of linking form over ℤ/p^{k} depending on whether θ is a quadratic residue (mod p) or not. The semi-group N_{p} is generated by the linking forms A^{k}(1) and A^{k}(θ), θ a quadratic non-residue where k ranges over the natural numbers. All

1. BILINEAR FORMS ON FINITE ABELIAN GROUPS 104

relations amongst the linking forms are generated by the following isometry

A^{k}(1) ⊕ A^{k}(1) ≅ A^{k}(θ) ⊕ A^{k}(θ).

Thus for odd primes the Grothendieck group of N_{p}, G(N_{p}), is isomorphic to the group ℤ^{∞} ⊕ (ℤ/2)^{∞}, where the infinite summands are generated by [A^{k}(1)] and the summands of order two are generated by [A^{k}(1)] − [A^{k}(θ)].

The main result of [KK] was the computation of the abelian semi-group of 2primary linking forms, N_{2}. The generators had already been discovered by Wall [Wa3] however the precise relations between them were unknown. We now review the generators of N_{2}. There are the cyclic linking forms A^{k}(n) for n odd. In this case A^{1}(n) = A^{1}(1) for all n, A^{2}(n) = A^{2}((−1)^{(n−1)/2}) and A^{k}(n) ≅ A^{k}(n′) if and only if n ≡ n′(8) for all k ≥ 3. There are also the hyperbolic and pseudo-hyperbolic linking forms with matrices

E^{k,0} = ( \begin{matrix}0 & 2^{−k} \\ 2^{−k} & 0 \end{matrix} ) (k ≥ 1) E^{k,1} = ( \begin{matrix}2^{1−k} & 2^{−k} \\ 2^{−k} & 2^{1−k} \end{matrix} ) (k ≥ 2).

Theorem 5.2 (Kawauchi and Kojima [KK] Theorem 0.1). N_{2} has a presentation with generators A^{k}(n), where n is 1(k = 1), ±1(k = 2), ±1 or ±5(k ≥ 3)), E^{k,0}(k ≥ 1) and E^{k,1}(k ≥ 2) and relations

\begin{matrix}A^{k}(n_{1}) ⊕ A^{k}(n_{2}) = A^{k}(n_{1} + 4) ⊕ A^{k}(n_{2} + 4) & (k ≥ 3) & (0.1) \\ A^{k}(n) ⊕ 2A^{k}(−n) = A^{k}(−n) ⊕ E^{k,0} & (k ≥ 1) & (0.2) \\ 3A^{k}(n) = A^{k}(−n + 4) ⊕ E^{k,0} & (k ≥ 2) & (0.3) \\ 2E^{k,0} = 2E^{k,1} & (k ≥ 2) & (0.4) \\ A^{k}(n_{1}) ⊕ A^{k+1}(n_{2}) = A^{k}(n_{1} + 2n_{2}) ⊕ A^{k+1}(n_{2} + 2n_{1}) & (k ≥ 1) & (1.1) \\ A^{k}(n) ⊕ E^{k+1,1} = A^{k}(n + 4) ⊕ E^{k+1,0} & (k ≥ 1) & (1.2) \\ E^{k,1} ⊕ A^{k+1}(n) = E^{k,0} ⊕ A^{k+1}(n + 4) & (k ≥ 2) & (1.3) \\ A^{k}(n_{1}) ⊕ A^{k+2}(n_{2}) = A^{k}(n_{1} + 4) ⊕ A^{k+2}(n_{2} + 4) & (k ≥ 1) & (2.1) \end{matrix}

1. BILINEAR FORMS ON FINITE ABELIAN GROUPS 105

We call a finite abelian group G, as opposed to a linking form, homogeneous if G is isomorphic to (ℤ/p^{k})^{l} for some integer l. It is implicit in the statement of the statement of Theorem 5.2 that every linking form b is isomorphic to a direct sum of linkings over homogeneous groups which is fact that was already proven in [Wa3, Lemma 8]. A complete list of homogeneous linking forms is given in [KK, 34-35]. Kawauchi and Kojima also provide a complete set of invariants which classify linking forms and we review these now. First let Ḡ_{p}^{k} be the subgroup of G generated by elements of order p^{s}, s ≤ k. Then let G̃_{p}^{k} := Ḡ_{p}^{k}/(Ḡ_{p}^{k−1} + pḠ_{p}^{k+1}). The group G̃_{p}^{k} is a vector space over ℤ/p and we define r_{p}^{k}(G) = dim_{ℤ/p}(G̃_{p}^{k}). Thus each linking form (G, b) is isomorphic to the direct sum ⨁_{k≥1}(G_{p}^{k}, b_{p}^{k}) with each G_{p}^{k} ≅ (ℤ/p^{k})^{r_{p}^{k}(G)}. The function

\begin{aligned}b̃_{p}^{k} : & G̃_{p}^{k} × G̃_{p}^{k} & → & ℚ/ℤ \\ & ([x], [y]) & ↦ & p^{k−1}b(x, y). \end{aligned}

is well defined and indeed defines a nonsingular linking form on G̃_{p}^{k}. For each linking form (G, b) we make the following

Definition 5.3 ([KK] Definition 1.1). Let c^{k}(b) ∈ G̃_{2}^{k} be the characteristic element of b̃_{2}^{k} which is uniquely defined by the identity

b̃_{2}^{k}(c^{k}(b), [x]) = b̃_{2}^{k}([x], [x]).

Observe that c^{k}(b) ≠ 0 if and only if (G, b) contains an indecomposable cyclic linking form A^{k}(n_{k}) as a summand. We now wrap up a lemma, corollary and definition of Kawauchi and Kojima into a single proposition.

Proposition 5.4 ([KK] §2). Let (G, b) be a linking form with c^{k}(b) = 0 and let q_{k} be the (well defined) function

\begin{aligned}q_{k} : & G_{2}/Ḡ_{2}^{k} & → & ℚ/ℤ \\ & [x] & ↦ & 2^{k−1}b(x, x). \end{aligned}

The complex number

GS_{k}(b) := ∑_{[x]∈G_{2}/Ḡ_{2}^{k}} exp(2πiq_{k}([x]))

(where exp denotes the standard exponential and i = \sqrt{−1}) is non-zero and there is an integer σ defined uniquely mod 8 such that GS_{k}(b) = |GS_{k}(b)| · exp(2πiσ/8). The invariant σ_{k}(b) is defined to be σ_{k}(b) ∈ ℤ/8 if c^{k}(b) = 0 and ∞ if c^{k}(b) ≠ 0.

Theorem 5.5 ([KK] 4.1). Two linking forms (G_{2}, b) and (G2^{′}, b′) are isometric if and only if r_{2}^{k}(G_{2}) = r^{k}(G′_{2}) and σ_{k}(b) = σ_{k}(b′) for all k ≥ 1.

Corollary 5.6. Let r_{p}^{k} : G(N_{2}) ⟶ ℤ be the kth-rank homomorphism defined on the Grothendieck group of N_{2} by

\begin{matrix}r_{2}^{k} : G(N_{2}) & ⟶ & ℤ[k] \\ ([b_{0}], [b_{1}]) & ↦ & r_{2}^{k}(b_{0}) − r_{2}^{k}(b_{1}). \end{matrix}

Then the homomorphism r_{2} which is the sum of r_{2}^{k} for all positive integers k gives an isomorphism

r_{2} : ⨁_{k=1}^{∞} r_{2}^{k} : G(N_{2}) ≅ ⨁_{k=1}^{∞} ℤ[k].

Proof. The homomorphism r_{2} is clearly surjective. Let b_{0} and b_{1} be linking forms for which ([b_{0}], [b_{1}]) lies in the kernel of r_{2}. If b is the linking form

b = ⨁_{{k:r_{2}^{k}(b_{0})≠0}} A^{k}(1)

then σ_{k}(b_{0} ⊕ b) = σ_{k}(b_{1} ⊕ b) = ∞ for every positive integer k. By Theorem 5.5, b_{0} ⊕ b ≅ b_{1} ⊕ b and hence the kernel of r_{2} is trivial.

2. QUADRATIC FORMS ON FINITE ABELIAN GROUPS 107

Remark 5.7. Notice that cancellation fails in N_{2} and hence information is lost in passing to G(N_{2}) ≅ (N_{2} ⊕ N_{2})/Δ, (where Δ = {([q], [q]) ∈ N_{2} × N_{2}}). In particular, if

Δ̄ = {([q_{0}], [q_{1}]) : (j[q_{0}], j[q_{1}]) ∈ Δ for some j ∈ ℤ^{+}},

then there is an isomorphism of sets Δ̄/Δ ≅ ℤ/2^{∞} ∪ ℤ/4^{∞}.

2. Quadratic forms on finite abelian groups

Recall from Section 9 that a homogeneous quadratic refinement of a linking form (G, b) is a function q : G → ℚ/ℤ such that q(−x) = q(x) for all x ∈ G and such that

b(x, y) = q(x + y) − q(x) − q(y) ∈ ℚ/ℤ

for all x, y ∈ G. We note that many authors including Wall consider quadratic refinements in ℚ/2ℤ in which case the defining equation is

q̄(x + y) = q̄(x) + q̄(y) + 2b(x, y) ∈ ℚ/2ℤ

where multiplication by 2 induces an isomorphism ·2 : ℚ/ℤ ≅ ℚ/2ℤ. The two definitions are equivalent with q̄ = (·2) ∘ q. We call any such q a quadratic linking form. Two quadratic linking forms (G_{0}, q_{0}) and (G_{1}, q_{1}) are isometric if there exists and isomorphism of abelian groups θ : G_{0} ≅ G_{1} such that q_{0} = q_{1} ∘ θ. Note that any isometry of quadratic refinements is necessarily an isometry of the associated bilinear forms. If G contains no 2-torsion then every bilinear form on G has a unique quadratic refinement and so we shall be concerned with bilinear forms on 2-groups. Given that Kawauchi and Kojima have classified bilinear forms on 2-groups we shall focus on the question of when two homogeneous quadratic refinements of the same linking form are isometric. Recall from Lemma 2.31 4, that if q is a fixed quadratic linking form refining b, then all other quadratic linking forms refining b are of the form q_{a}, where

2. QUADRATIC FORMS ON FINITE ABELIAN GROUPS 108

a ∈ G and 2a = 0. Thus our goal can be restated as follows: given a, c ∈ Ḡ_{2}^{1} when is q_{a} ≅ q_{c}? We shall discover a most satisfactory answer to this question in Corollary 5.13. We turn now to the definition of the invariants which resolve this question. Recall Wall’s theorem that every quadratic form (G, q) is the boundary of some even quadratic form κ(H, λ, 0),

0 ⟶ H ⟶^{λ̂} H^{∗} → G ⟶ 0.

Lemma 5.8. Let q be a quadratic linking form and κ(λ, 0) any nondegenerate even quadratic function such that q^{ev}(κ) = q. The signature of λ (mod 8) is independent of the choice of κ(λ, 0) and is thus an invariant of q:

σ(q) := σ(λ) (mod 8) ∈ ℤ/8.

Proof. We use the technique of algebraic gluing from Lemma 3.6. Let κ_{0} = κ(λ_{0}) and κ_{1} = κ(λ_{1}) be two nondegenerate quadratic forms inducing q, then the quadratic form (κ_{0}∪_{q} κ_{1}^{−}) defined in Lemma 3.6 is nonsingular and even and hence has signature divisible by 8 according to [MH, Theorem 5.1]. If λ is the bilinear form associated to (κ_{0} ∪_{q} κ_{1}^{−}) then λ ⊗ Id_{ℚ} = (λ_{0} ⊗ Id_{ℚ}) ⊕ (−λ_{1} ⊗ Id_{ℚ}) and hence σ(λ) = σ(λ_{0}) − σ(λ_{1}) and σ(q) is well-defined.

Theorem 5.9 (Nikulin [Ni]). Two quadratic linking forms q and q′ are isometric if and only if b(q) ≅ b(q′) and σ(q) ≡ σ(q′)(mod 8).

The invariant σ(q) may also be computed by a Gauss-sum formula as follows.

Definition 5.10. Let (G, q) be a quadratic linking form on a finite abelian group. Define the Gauss sum invariant of q by

GS(q) = 1/\sqrt{|G|} · ∑_{x∈G} exp(2πiq(x)) ∈ ℂ.

2. QUADRATIC FORMS ON FINITE ABELIAN GROUPS 109

Theorem 5.11 (Milgram [Milg]). Let (G, q) be a quadratic linking form and let κ(λ, 0) be a nondegenerate quadratic function such that q^{ev}(κ) = q. Then

GS(q) = exp(2πiσ(λ)/8).

Theorem 5.11 gives another proof that σ(q) := σ(λ) (mod 8) is well defined. It follows in particular from Theorem 5.11, that GS(q) is never zero. Moreover, the important part of this complex number for our purposes is its argument, so we make the following

Definition 5.12. Let (G, q) be a quadratic linking form. Define the Kervaire-Arf invariant of q, K(q) ∈ ℚ/ℤ, to be the argument of GS(q) divided by 2π.

Corollary 5.13. Two quadratic linking forms q and q′ are isometric if and only if b(q) ≅ b(q′) and K(q) = K(q′). For a fixed quadratic linking form q and for any a, c ∈ Ḡ_{2}^{1}, q_{a} and q_{c} are isometric if and only if q(a) = q(c).

Proof. The first statement is a trivial consequence of Theorems 5.9 and 5.11. The second statement follows from the following simple calculation.

GS(q_{a}) = ∑_{x∈G} exp(2πiq_{a}(x))
= ∑_{x∈G} exp(2πi(q(x + a) − q(a)))
= [∑_{x∈G} exp(2πiq(x))] .exp(−2πiq(a))
= GS(q).exp(−2πiq(a)).

Thus K(q_{a}) = K(q) − q(a) and so q_{a} ≅ q_{c} if and only if and only if q(a) = q(c). We shall not give Nikulin’s proof of Theorem 5.9 here and mention only that it is achieved by induction over the number of generators of G and the classification of bilinear forms on 2-groups (which Nikulin seems to have obtained independently from Kawauchi and Kojima). To conclude this section we mention another result of Wall’s which reduces the problem of classifying quadratic forms to the classification

2. QUADRATIC FORMS ON FINITE ABELIAN GROUPS 110

of bilinear forms. If G is a finite 2-group choose a basis for G, that is a set of generators {x_{i}} for cyclic groups of which G is a direct sum. Let G_{×2} be the group with generators {x′_{i}} where the order of x′_{i} is twice the order of x_{i}. There is a short exact sequence of groups

0 → Ḡ_{×2}^{1} → G_{×2} → G → 0.

Theorem 5.14 (Wall [Wa3] Theorem 5). A quadratic linking form q : G → ℚ/ℤ determines and is determined by a linking form B(q) : G_{×2} × G_{×2} → ℚ/ℤ.

With this result of Wall it is not difficult to use the classification of Kawauchi and Kojima to prove Nikulin’s theorem. In fact, let b(q) : G × G → ℚ/ℤ be the bilinear form which q refines. By definition GS_{k}(B(q)) = GS_{k−1}(b(q)) for k > 1 and GS_{1}(B(q)) = GS(q). (Note that since G_{×2} has no elements of order two c^{1}(B(q)) = 0 and the results of [KK] give an alternate proof that GS(q) is non-zero.) Theorem 5.9 now follows from Theorems 5.14 and 5.5 applied to B(q) and the fact that linking forms on groups of odd order have unique quadratic refinements. In fact we may now state the following extension of Theorem 5.5 to quadratic linking forms.

Corollary 5.15. Two quadratic linking forms (G_{2}, q) and (G2^{′}, q′) are isometric if and only if r_{2}^{k}(G_{2}) = r^{k}(G′_{2}) and σ_{k}(B(q)) = σ_{k}(B(q′)) for all k ≥ 1.

Proof. By Theorem 5.14 q and q′ are isometric if and only if B(q) and B(q′) are isometric and this occurs according to Theorem 5.5 if and only if r_{2}^{k}(G_{2}) = r^{k}(G′_{2}) and σ_{k}(B(q)) = σ_{k}(B(q′)) for all k ≥ 1.

Remark 5.16. Let q : G → ℚ/ℤ be a quadratic linking form defined on the 2-group G and let the range of q, R(q), be the images of q in ℚ/ℤ counted with multiplicity. It follows from Corollary 5.15 that if (G, q′) is another quadratic liking form then q is isometric to q′ if and only if R(q) = R(q′).

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3. Quadratic functions on finite abelian groups

A quadratic linking functions is a function q : G → ℚ/ℤ which refines a bilinear form b but which need not be homogeneous. Just as in the homogeneous case we may define the Gauss-sum invariant and the associated Kervaire-Arf invariant.

GS(q) = ∑_{x∈G} exp(2πiq(x)) ∈ ℂ K(q) = Arg(GS(q))/2π

However the signature invariant of a general quadratic linking function must be somewhat modified as follows. Recall that for every quadratic linking function q, Corollary 2.36 guarantees a characteristic quadratic function κ such that q^{c}(κ) ≅ q.

Proposition 5.17. Let (G, q) be a quadratic linking function and let κ_{0} = κ(λ_{0}, α_{0}) be a nondegenerate quadratic function such that δ^{c}(κ_{0}) = q. Then the quantity

s̄_{1}(q) := (λ_{0}^{−1}(α_{0}, α_{0}) − σ(λ_{0}))/8 ∈ ℚ/ℤ.

is independent of the choice of κ_{0}.

Proof. Suppose that κ_{0} and κ_{1} are two quadratic functions such that δ^{c}(κ_{0}) = q = δ^{c}(κ_{1}). Then by Lemma 3.6 we may form the nonsingular characteristic quadratic function κ_{0} ∪_{Id} κ_{1}^{−}. We let λ and α denote respectively the bilinear and linear parts of κ_{0} ∪_{Id} κ_{1}^{−}. Since α is characteristic for λ it is well known that λ(α, α) ≡ σ(λ) (mod 8) (see for example [MH]). However [λ(α, α) − σ(λ)]/8 mod ℤ = [λ_{0}^{−1}(α_{0}, α_{0}) − λ_{1}^{−1}(α_{1}, α_{1}) − (σ(λ_{0}) − σ(λ_{1}))]/8 mod ℤ

0 mod ℤ = [λ_{0}^{−1}(α_{0}, α_{0}) − σ(λ_{0})]/8 − [λ_{1}^{−1}(α_{1}, α_{1}) − σ(λ_{1})]/8 mod ℤ. As the right hand side is the difference of the possible values for s̄_{1}(q) we have proven the proposition.

Remark 5.18. We have called this invariant of a quadratic linking functions s̄_{1} because when q = δ^{c}(κ_{0}) then s̄_{1}(q) = s̄_{1}(∂L(κ_{0})) where s̄_{1} is the almost diffeomorphism invariant of ∂L_{0} which was defined in Section 2.12.

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Proposition 5.19. Let q : G → ℚ/ℤ be a quadratic linking function on a finite abelian group. Then

s̄_{1}(q) = −K(q)

Proof. By the proof of Corollary 2.36 there is a nondegenerate characteristic quadratic function κ = κ(H, λ, α) with λ an even bilinear form such that q = δ^{c}(κ). It follows that 2 divides α and so we let a = [α/2] ∈ G = H^{∗}/λ̂(H). Moreover, if we let q^{o} = δ^{c}(κ(λ, 0)) be the quadratic linking form defined by λ alone then q = q_{a}^{o} by Lemma 2.37. By Proposition 5.17

s̄_{1}(q) = [λ^{−1}(α, α) − σ(λ)]/8 mod ℤ
= λ^{−1}(α/2, α/2)/2 − K(q^{o})
= q′(a) − K(q^{o})
= −K(q).

The last step follows by the arithmetic of Corollary 5.13 which showed that K(q_{a}^{o}) = K(q^{o}) − q^{o}(a) (and which made no use of the fact operating then that 2a = 0).

Corollary 5.20. If q_{0} and q_{0} are two quadratic linking functions, then K(q_{0} ⊕ q_{1}) = K(q_{0}) + K(q_{1}).

Proof. This will follow from the simple fact that s̄_{1} is additive. For let κ_{0}(λ_{0}, α_{0}) and κ_{1}(λ_{1}, α_{1}) be two quadratic functions such that q_{i} = δ^{c}(κ_{i}). Then

s̄_{1}(q_{0} ⊕ q_{1}) = [(λ_{0} ⊕ λ_{1})^{−1}(α_{0}, α_{1}) − σ(λ_{0} ⊕ λ_{1})]/8 mod ℤ
= [λ_{0}^{−1}(α_{0}, α_{0}) − σ(λ_{0})]/8 + [λ_{1}^{−1}(α_{1}, α_{1}) − σ(λ_{1})]/8 mod ℤ
= s̄_{1}(q_{0}) + s̄_{1}(q_{1}))

Now recall the Wilkens data (b(q), β(q)) of a quadratic linking function q which is the pair composed of the linking form refined by q and the even element β(q) defined in 2.31, 5. The element β(q) is twice the linear part of q.

Lemma 5.21. Let q : G → ℚ/ℤ be a quadratic linking function. If the order of G is odd then (b(q), β(q)) is a complete invariant of q.

Proof. In a group of odd order there is a unique quadratic linking form q^{o} refining b(q). As there is a unique γ ∈ G such that q = q_{γ}^{o} and γ = (1/2)β(q) is determined by β(q), we see that (b(q), β(q)) determines q.

Before stating and proving the classification of quadratic linking functions we establish notation for quadratic linking functions parallel to the notation for linking forms in Section 1. Let q^{k}(n), q^{k,0} and q^{k,1} denote homogeneous quadratic refinements of A^{k}(n), E^{k,0} and E^{k,1} respectively. We define homogeneous quadratic refinements q^{k}(n), q^{k,0}, and q^{k,1} of A^{k}(n), E^{k,0}, and E^{k,1} as follows. For r, s ∈ ℤ,

\begin{matrix}(k ≥ 1) A^{k}(n) = (ℤ/2^{k}[e], A_{n}^{k}(e, e) = n2^{−k}) \\ q^{k}(n)(re) = r^{2}n2^{−k−1} \end{matrix}

\begin{matrix}(k ≥ 1) E^{k,0} = (ℤ/2^{k}[e_{1}] ⊕ ℤ/2^{k}[e_{2}], E^{k,0}(e_{i}, e_{i}) = 0, E^{k,0}(e_{1}, e_{2}) = 2^{−k}) \\ q^{k,0}(re_{1} + se_{2}) = rs2^{−k} \end{matrix}

\begin{matrix}(k ≥ 2) E^{k,1} = (ℤ/2^{k}[e_{1}] ⊕ ℤ/2^{k}[e_{2}], E^{k,1}(e_{i}, e_{i}) = 2^{1−k}, E^{k,1}(e_{1}, e_{2}) = 2^{−k}) \\ q^{k,1}(re_{1} + se_{2}) = (r^{2} + rs + s^{2})2^{−k}. \end{matrix}

We shall continue to with our convention of using a subscript to denote a linear perturbation of a quadratic function, q_{a}(x) = q(x) + b(x, a). So for example

[q^{k}(n)_{e}](re) = q^{k}(n)(re) + A^{k}(n)(re, e) = r^{2}n2^{−k−1} + rn2^{−k}.

Theorem 5.22. Let q_{0} and q_{1} be quadratic linking functions over a finite abelian group G. Then q_{0} is isometric to q_{1} if and only if (b(q_{0}), β(q_{0})) ≅ (b(q_{1}), β(q_{1})) and K(q_{0}) = K(q_{1}).

Proof. By Lemma 5.21, (b(q), β(q)) is an invariant of the isometry class of any quadratic linking function q. Moreover, K(q) is manifestly an invariant. Therefore,

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it remains to show that q_{0} and q_{1} are isometric if (b(q_{0}), β(q_{0})) ≅ (b(q_{1}), β(q_{1})) and K(q_{0}) = K(q_{1}). By Lemma 5.21 (or by the results of Wilkens’ thesis) the theorem is true if b(q_{0}) and b(q_{1}) are linking forms on a group of odd order. We therefore assume that the linking forms are defined on 2-primary groups. Let q have Wilkens data (b(q), β(q)) and a fixed Kervaire-Arf invariant K(q) ∈ ℚ/ℤ. The strategy of the proof shall be to show that q is uniquely determined up to isometry by these data using induction on the number of indecomposable summands in b(q).

We first observe that if θ : (b(q_{0}), β_{0})) ≅ (b(q_{1}), β(q_{1})) then we may replace (b(q_{1}), β(q_{1})) with (b(q_{1}) ∘ (θ × θ), θ^{−1}(β(q_{1})). We shall therefore always assume that (b(q_{0}), β(q_{0})) = (b(q_{1}), β(q_{1})). In order to prove the existence of the isometries of quadratic linking functions predicted by the theorem we shall constantly use the following lemma.

Lemma 5.23. Two quadratic linking functions (G_{0}, q_{0}) and (G_{1}, q_{1}) are isometric if and only if there are bases {e_{1}, …, e_{n}} and {f_{1}, …, f_{n}} for G_{0} and G_{1} respectively such that and q_{0}(e_{i}) = q_{1}(f_{i}) and b(q_{0})(e_{i}, e_{j}) = b(q_{1})(f_{i}, f_{j}) for all i and j.

Proof of Lemma. Clearly the condition given is necessary for q_{0} and q_{1} to be isometric. It is sufficient because the relation q_{i}(x + y) = q_{i}(x) + q_{i}(y) + b_{i}(x, y), i = 0, 1, means that q_{i} is determined by b(q_{i}) and the values q_{i} takes on a basis.

The base case of our induction is when b(q) is indecomposable. In this case Wilkens’ thesis implies that there are at most two quadratic linking functions with invariants (G, b, β) and one may check using Wilkens’ calculations that in the cases where there are two quadratic linking functions then they have distinct Kerviare-Arf invariant. We now give a proof of the theorem in the indecomposable case which does not rely on Wilkens’ results. Consider the case where (b(q), β(q)) = (A^{k}(n), β) is a linking form on the cyclic group ℤ/2^{k}. We must therefore consider q^{k}(n)_{a} and

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q^{k}(n)_{a+2^{k−1}e} where 2a = β. We have already noted that the proof of Corollary 5.13 shows that K(q_{a}) = K(q) − q(a) for any quadratic linking function. Hence

K(q^{k}(n)_{a+2^{k−1}e}) − K(q^{k}(n)_{a}) = q^{k}(n)(a + 2^{k−1}e) − q^{k}(n)(a)
= (na^{2} + 2^{k}na + n2^{2k−2} − na^{2}).2^{−k−1}
= na2^{−1} + n2^{k−3}.

We see that K(q^{k}(n)_{a+2^{k−1}e}) ≠ K(q^{k}(n)_{a} when k = 1, when k = 2 and 2|a and when k ≥ 3 and 2 |̸ a (recall that n is odd). In these cases the two quadratic refinements are not isometric, this is detected by the Kervaire-Arf invariant and there is nothing further to prove. In the remaining cases (k = 2 and 2 |̸ a or k ≥ 3 and 2|a) we must show that q^{k}(n)_{a+2^{k−1}e} and q^{k}(n)_{a} are isometric. We let θ : ℤ/2^{k} → ℤ/2^{k} be multiplication by (1+2^{k−1}) and verify that θ is the desired isometry. Firstly let k = 2 and 2 |̸ a.

q^{2}(n)_{a}(3e) = q^{2}(n)(e + 2e) + A^{2}(n)(e + 2e, a)
= q^{2}(n)(e) + q^{2}(2e) + A^{2}(n)(e, 2e) + A^{2}(n)(2e, a) + A^{2}(n)(e, a)
= q^{2}(n)(e) + 2^{−1} + A^{2}(n)(e, 2e) + 2^{−1} + A^{2}(n)(e, a)
= q^{2}(n)(e) + A^{2}(n)(e, a + 2e)
= q^{2}(n)_{a+2e}(e)

Hence by Lemma 5.23 applied to (q^{k}(n)_{a+2^{k−1}e}, e) and (q^{k}(n)_{a}, 3e) we have the desired isometry. In the second case k ≥ 3 and 2|a.

q^{k}(n)_{a}((1 + 2^{k−1})e) = q^{k}(n)(e + 2^{k−1}e) + A^{k}(n)(e + 2^{k−1}e, a)

= q^{k}(n)(e) + q^{k}(2^{k−1}e) + A^{k}(n)(e, 2^{k−1}e) + A^{k}(n)(a, e + 2^{k−1}e)

= q^{k}(n)(e) + 0 + A^{k}(n)(e, 2^{k−1}e) + A^{k}(n)(a, e)
= q^{k}(n)(e) + A^{k}(n)(e, a + 2^{k−1}e)
= q^{k}(n)_{a+2^{k−1}e}(e)

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Again by Lemma 5.23 applied to (q^{k}(n)_{a+2^{k−1}e}, e) and (q^{k}(n)_{a}, (1 + 2^{k})e) we have the desired isometry. We now consider the hyperbolic and pseudo-hyperbolic linking forms. When (b(q), β(q)) = (E_{0}^{1}, 0) the classification of quadratic refinements via the Kervaire-Arf invariant is well known: K(q^{1}) = K(q_{e_{1}}^{1}) = K(q_{e_{2}}^{1}) = 0 and K(q_{e_{1}+e_{2}}^{1}) = 1/2. It is perhaps most instructive simply to list the table of values for all four quadratic refinements of E^{1}.

q^{1} q_{e_{1}}^{1} q_{e_{2}}^{1} q_{e_{1}+e_{2}}^{1}
0 0 0 0 0
e_{1}  0 0 1/2 1/2
e_{2}  0 1/2 0 1/2
e_{1} + e_{2} 1/2 0 0 1/2

The first three quadratic linking functions are evidently isometric via permutations of e_{1}, e_{2} and e_{1} + e_{2}. We now consider Wilkens data (b(q), β(q)) = (E^{k,ϵ}, β) where ϵ = 0, 1, k ≥ 2, β = 2a and a = a_{1}e_{1} + a_{2}e_{2} is any element in ℤ/2^{k} ⊕ ℤ/2^{k}. We let c = c_{1}e_{1} + c_{2}e_{2} denote any of the elements e_{1}, e_{2} or e_{1} + e_{2} so that c_{i} = 0 or 1. Again we apply Corollary 5.13 to calculate that

K(q_{a+2^{k−1}c}^{k,ϵ}) − K(q_{a}^{k,ϵ}) = q^{k,ϵ}(a + 2^{k−1}c) − q^{k,ϵ}(a)

= q^{k,ϵ}(a) + q^{k,ϵ}(2^{k−1}c) + E^{k,ϵ}(a, 2^{k−1}c) − q^{k,ϵ}(a) = E^{k,ϵ}(a, 2^{k−1}c) = (a_{1}c_{1} + a_{2}c_{2})2^{−1}

= \{ \begin{matrix}0 & if (a_{1}c_{1} + a_{2}c_{2})is even \\ 1/2 & if (a_{1}c_{1} + a_{2}c_{2})is odd \end{matrix} \} .

Hence we see Wilkens’ criteria that when the order of the group is greater than 4, (k ≥ 2), that (E^{k,ϵ}, β) is ambiguous if and only if 4 does not divide β = 2a_{1}e_{1}+2a_{2}e_{1}. More importantly we must exhibit an isometry θ from q_{a}^{k,ϵ} to q_{a+2^{k−1}f}^{k,ϵ} when a_{1}c_{1}+a_{2}c_{2} is odd. Now either both a_{i} are odd in which case we assume (as we may interchange

3. QUADRATIC FUNCTIONS ON FINITE ABELIAN GROUPS 117

bases elements at will) that c_{2} = 0 or precisely one a_{i} is odd in which case we assume it to be a_{1} and hence c_{1} = 1. We describe the necessary isometry θ in these two cases and present the matrices which by Lemma 5.23 ensures that θ is an isometry.

(a_{i} odd, c_{1} = 1, c_{2} = 0) \{ \begin{matrix}q_{a}^{k,ϵ} & →^{θ} & q_{a+2^{k−1}f}^{k,ϵ} \\ e_{1} & ↦ & e_{1} \\ e_{2} & ↦ & e_{2} + 2^{k−1}e_{1}. \end{matrix} \}

[ \begin{matrix}q_{a}^{k,ϵ}(e_{1}) & E^{k,ϵ}(e_{1}, e_{2}) \\ ∗ & q_{a}^{k,ϵ}(e_{2}) \end{matrix} ] = [ \begin{matrix}q_{a}^{k,ϵ}(e_{1}) & 2^{−k} \\ ∗ & q_{a}^{k,ϵ}(e_{2}) \end{matrix} ]

[ \begin{matrix}q_{a+2^{k−1}c}^{k,ϵ}(e_{1}) & E^{k,ϵ}(e_{1}, e_{2} + 2^{k−1}e_{1}) \\ ∗ & q_{a+2^{k−1}c}^{k,ϵ}(e_{2} + 2^{k−1}e_{1}) \end{matrix} ] = [ \begin{matrix}q_{a}^{k,ϵ}(e_{1}) + c_{2} & 2^{−k} \\ ∗ & q_{a}^{k,ϵ}(e_{2}) + 2^{−1}(c_{1} + a_{2}) \end{matrix} ]

(a_{1} odd, a_{2} even, c_{1} = 1) \{ \begin{matrix}q_{a}^{k,ϵ} & →^{θ} & q_{a+2^{k−1}f}^{k,ϵ} \\ e_{1} & ↦ & e_{1} + 2^{k−1}c_{2}e_{2} \\ e_{2} & ↦ & e_{2} + 2^{k−1}e_{2}. \end{matrix} \}

[ \begin{matrix}q_{a+2^{k−1}c}^{k,ϵ}(e_{1} + 2^{k−1}c_{2}e_{2}) & E^{k,ϵ}(e_{1}, e_{2} + 2^{k−1}c_{2}e_{1}) \\ ∗ & q_{a+2^{k−1}c}^{k,ϵ}(e_{2} + 2^{k−1}e_{1}) \end{matrix} ] =

[ \begin{matrix}q_{a}^{k,ϵ}(e_{1}) + 2^{−1}(c_{2} + c_{2}a_{1}) & 2^{−k} \\ ∗ & q_{a}^{k}(e_{2}) + 2^{−1}(c_{1} + a_{1}) \end{matrix} ]

This completes the classification of quadratic linking forms refining indecomposable pairs (b(q), β). We call the Wilkens data (b, β) ambiguous if there is more than one isometry class of quadratic linking function with Wilkens data (b, β). We call the Wilkens data (b, β) unambiguous if there is a unique quadratic linking function with invariants (b, β). We have just reproven Wilkens’ result that an indecomposable pair (b, β) is ambiguous if and only if the order of the group is less than or equal to 4 and β = 0 or the order of the group is greater than 4 and β is not divisible by 4. Moreover

3. QUADRATIC FUNCTIONS ON FINITE ABELIAN GROUPS 118

we note that in all ambiguous cases except for (A^{1}(1), 0) that the K(q_{0})−K(q_{1}) = 1/2 where q_{0} and q_{1} are the two possible refinements. When (b(q), β(q)) = (A^{1}(1), 0) then there are two possible quadratic refinements q^{1}(1) and q^{1}(1)_{e} and these have distinct Kervaire-Arf invariants:

K(q^{1}(1)) = \frac{1}{8} and K(q^{1}(1)_{e}) = \frac{7}{8}.

The inductive step requires us to verify the theorem in a large number of special cases. We start with a simple case by way of illustration. Suppose that (b(q), β(q)) = (b_{0}, β_{0}) ⊕ (b′, β′) where (b_{0}, β_{0}) is an unambiguous pair with quadratic refinement q_{0}. It follows that q ≅ q_{0} ⊕ q′ where q′ refines (b′, β′). But since the Kervaire-Arf invariant is additive over block sum K(q′) = K(q) − K(q_{0}) is determined and hence by induction q′ is uniquely determined. If suffices therefore to restrict our attention to pairs (b(q), β(q)) consisting entirely of ambiguous irreducible components. If these all have β(q) = 0 then all the quadratic linking functions are homogeneous and hence quadratic linking forms in which case Nikulin has proven the result. We thus check the cases in which at least one of the ambiguous indecomposable pairs has β(q) ≠ 0 and therefore 4 |̸ β(q). We label that the ambiguous indecomposable inhomogeneous Wilkens data (I, β) and recall that they are as follows:

(A^{k}(n), β) k > 2, 4 |̸ β, (E^{k,ϵ}, β) k > 1, 4 |̸ β.

Suppose that (b(q), β(q)) = (A^{1}(1), 0) ⊕ (I, β) then there are four possible quadratic refinements and four different values of K(q) so the result is proved. In the remaining cases we consider Wilkens data of the form (I, β_{0}) ⊕ (J, β_{1}) ⊕ (b′, β′) where (I, β_{0}) is ambiguous, indecomposable and inhomogeneous and (J, β_{1}) is ambiguous and either indecomposable or (A^{1}(1)⊕A^{1}(1), 0). Let q^{I} and ∗q^{I} denote the two isometry classes of quadratic refinements of (I, β_{0}) and let q^{J} and ∗q^{J} be quadratic refinements of (J, β_{1}) such that K(q^{J}) − K(∗q^{J}) = 1/2. We shall show that ∗q^{I} ⊕ ∗q^{J} ≅ q^{I} ⊕ q^{J}. It follows

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that every quadratic refinement of (I, β_{0}) ⊕ (J, β_{1}) ⊕ (b′, β′) is isometric to q^{I} ⊕ q′′ for some q′′ which refines (J, β_{1}) ⊕ (b′, β′) and hence by induction that the value of K(q) determines the isometry class of any quadratic refinement of (I, β_{0})⊕(J, β_{1})⊕(b′, β′).

We now proceed to the necessary computations. Let a_{i} be elements of the domain of I or J such that 2a_{i} = β_{i}. From our computations in the indecomposable case we know that if I or J is isomorphic to E^{k,ϵ} with k > 1 then, by swapping e_{1} and e_{2} if necessary, we may assume that K(q_{a_{i}}^{k,ϵ}) − K(q_{a_{i}+2^{k−1}e_{1}}^{k,ϵ}) = 1/2. Finally, when considering pairs of similar linking forms we shall distinguish generators by the place they hold. So, for example, we shall denote generators of A^{k}(n_{0}) ⊕ A^{l}(n_{1}) by (e, 0) and (0, e).

(b(q), β(q)) = (A^{k}(n_{0}), β_{1}) ⊕ (A^{l}(n_{1}), β_{2}) (k ≥ 3, l ≥ 2)

\begin{matrix}q^{k}(n_{0})_{a_{0}} & ⊕ & q^{l}(n_{1})_{a_{1}} & → & q^{k}(n_{0})_{a_{0}+2^{k−1}e} & ⊕ & q^{l}(n_{1})_{a_{1}+2^{l−1}e} \\ (e & , & 0) & ↦ & (e & , & 2^{l−1}e) \\ (0 & , & e) & ↦ & (2^{k−1}e & , & e) \end{matrix}

(b(q), β(q)) = (A^{k}(n_{0}), β_{1}) ⊕ (E^{l,ϵ}, β_{2}) (k, l ≥ 2).

\begin{matrix}q^{k}(n_{0})_{a_{0}} & ⊕ & q_{a_{1}}^{l,ϵ} & → & q^{k}(n_{0})_{a_{0}+2^{k−1}e} & ⊕ & q_{a_{1}+2^{l−1}e_{1}}^{l,ϵ} \\ (e & , & 0) & ↦ & (e & , & 2^{l−1}e_{1}) \\ (0 & , & e_{1}) & ↦ & (0 & , & e_{1}) \\ (0 & , & e_{2}) & ↦ & (2^{k−1}e & , & e_{2}) \end{matrix}

(b(q), β(q)) = (E^{k,ϵ}, β_{1}) ⊕ (E^{l,ϵ}, β_{2})) (k, l ≥ 2).

\begin{matrix}q_{a_{0}}^{k,ϵ} & ⊕ & q_{a_{1}}^{l,ϵ} & → & q_{a_{0}+2^{k−1}e}^{k,ϵ} & ⊕ & q_{a_{1}+2^{l−1}e_{1}}^{l,ϵ} \\ (e_{1} & , & 0) & ↦ & (e_{1} & , & 0) \\ (e_{2} & , & 0) & ↦ & (e_{2} & , & 2^{l−1}e_{1}) \\ (0 & , & e_{1}) & ↦ & (0 & , & e_{1}) \\ (0 & , & e_{2}) & ↦ & (2^{k−1}e_{1} & , & e_{2}) \end{matrix}

(b(q), β(q)) = (A^{k}(n_{0}), β_{1}) ⊕ (E^{1,0}, 0) (k ≥ 3, 4 |̸ β_{1}).

\begin{matrix}q^{k}(n_{0})_{a_{0}} & ⊕ & q_{e_{1}+e_{2}}^{1} & → & q^{k}(n_{0})_{a_{0}+2^{k−1}e} & ⊕ & q^{1} \\ (e & , & 0) & ↦ & (e & , & e_{1} + e_{2}) \\ (0 & , & e_{1}) & ↦ & (2^{k−1}e & , & e_{1}) \\ (0 & , & e_{2}) & ↦ & (2^{k−1}e & , & e_{2}) \end{matrix}

(b(q), β(q)) = (A^{k}(n_{0}), β_{1}) ⊕ (A^{1}(1) ⊕ A^{1}(1), 0)) (k ≥ 3, 4 |̸ β_{1}).

\begin{matrix}q^{k}(n_{0})_{a_{0}} & ⊕ & q^{1}(1)_{a_{1}} & ⊕ & q^{1}(1)_{a_{2}} & → & q^{k}(n_{0})_{a_{0}+2^{k−1}e} & ⊕ & q^{1}(1)_{a_{1}+e} & ⊕ & q^{1}(1)_{a_{2}+e} \\ (e & , & 0 & , & 0) & ↦ & (e & , & e & , & e) \\ (0 & , & e & , & 0) & ↦ & (2^{k−1}e & , & e & , & 0) \\ (0 & , & 0 & , & e) & ↦ & (2^{k−1}e & , & 0 & , & e) \end{matrix}

(b(q), β(q)) = (E^{k,ϵ}, β_{1}) ⊕ (E^{1,0}, 0)) (k ≥ 2, 4 |̸ β_{1}).

\begin{matrix}q_{a_{0}}^{k,ϵ} & ⊕ & q_{e_{1}+e_{2}}^{1} & → & q_{a_{0}+2^{k−1}e_{1}}^{k,ϵ} & ⊕ & q^{1} \\ (e_{1} & , & 0) & ↦ & (e_{1} & , & 0) \\ (e_{2} & , & 0) & ↦ & (e_{2} & , & e_{1} + e_{2}) \\ (0 & , & e_{1}) & ↦ & (2^{k−1}e_{1} & , & e_{1}) \\ (0 & , & e_{2}) & ↦ & (2^{k−1}e_{1} & , & e_{2}) \end{matrix}

(b(q), β(q)) = (E^{k,ϵ}, β_{1}) ⊕ (A^{1}(1) ⊕ A^{1}(1), 0)) (k ≥ 2, 4 |̸ β_{1}).

\begin{matrix}q_{a_{0}}^{k,ϵ} & ⊕ & q_{a_{1}}^{1} & → & q_{a_{0}+2^{k−1}e}^{k,ϵ} & ⊕ & q_{a_{1}+2^{l−1}e_{1}}^{1} \\ (e_{1} & , & 0) & ↦ & (e_{1} & , & 0) \\ (e_{2} & , & 0) & ↦ & (e_{2} & , & 2^{l−1}e_{1}) \\ (0 & , & e_{1}) & ↦ & (0 & , & e_{1}) \\ (0 & , & e_{2}) & ↦ & (2^{k−1}e_{1} & , & e_{2}) \end{matrix}

This concludes the proof of Theorem 5.22.

Corollary 5.24. Let P_{0} and P_{1} be highly connected rational homology spheres of dimension 7 or 15 with isometric Wilkens invariants (G_{0}, b_{0}, β_{0}) and (G_{1}, b_{1}, β_{1}). Then P_{0} and P_{1} are almost diffeomorphic (resp. diffeomorphic) if and only if s̄_{1}(P_{0}) = s̄_{1}(P_{1}) (resp. s_{1}(P_{0}) = s_{1}(P_{1}) and Σ_{P_{0}} = Σ_{P_{1}}).

Proof. By Theorem B, P_{0} and P_{1} are almost diffeomorphic if and only if Q(P_{0}) = (G_{0}, q_{0}^{c}, β_{0}) is isometric to Q(P_{1}) = (G_{1}, q_{1}^{c}, β_{1}). As P_{0} and P_{1} are rational homology spheres q_{0}^{c} = q_{0} and q_{1}^{c} = q_{1} are quadratic linking functions which by hypothesis define isometric Wilkens data. Now by Theorem 5.22, q_{0} and q_{1} are isometric if and only if K(q_{0}) = K(q_{1}) but by Proposition 5.19, K(q_{i}) = −s̄_{1}(q_{i}), i = 0, 1, and by Remark 5.18, s̄_{1}(q_{i}) = s̄_{1}(P_{i}). Hence P_{0} and P_{1} are almost diffeomorphic if and only

if s̄_{1}(P_{0}) = s̄_{1}(P_{1}). By Theorem 4.9, P_{0} and P_{1} are diffeomorphic if and only if q_{0} is

isometric to q_{1}, s_{1}(P_{0}) = s_{1}(P_{1}) and Σ_{P_{0}} = Σ_{P_{1}}. As s̄_{1} is a multiple of s_{1}, we conclude

that P_{0} and P_{1} are diffeomorphic if and only if s_{1}(P_{0}) = s_{1}(P_{1}) and Σ_{P_{0}} = Σ_{P_{1}}.

CHAPTER 6 Classification up to homeomorphism and homotopy

In this final chapter we shall restrict our attention to highly connected manifolds of dimension 7 and thus the pair (L, P) shall always denote an 8-dimensional handlebody L with boundary P. We remind the reader that we consider only smooth manifolds P. We shall extend the almost diffeomorphism classification obtained in Chapter 4 to a classification up to P L-homeomorphism, homeomorphism and homotopy. The classification of manifolds is often presented in the opposite direction, moving from the homotopy classification to the homeomorphism classification via the surgery exact sequence and then upwards via smoothing theory to the P L-classification and the smooth classification. However, in the case of highly connected manifolds it has proven easier to classify up to almost diffeomorphism first and then descend to the other categories using smoothing theory and the surgery exact sequence in the opposite direction.

1. Smoothing theory

We begin by briefly recalling the theory of smoothing manifolds. Suppose that M is a P L manifold of dimension greater than 4. If we embed M in Euclidean space of sufficiently highly dimension then the embedding we choose is unique up to isotopy and hence M has a well defined stable normal bundle which we denote by ν_{M}^{P L} : M → BP L. Now there is a forgetful map from π_{O,P L} : BO → BP L which classifies the universal smooth bundle γ : EO → BO as a P L-bundle. It is a remarkable fact (proven in [HM]) that the question of whether M may be equipped

1. SMOOTHING THEORY 124

with a smooth structure reduces to the question of whether the map stable P L-normal bundle of M lifts through π_{O,P L}.

[figure: BO; M; BP L; π_{O,P L}; ν_{M}^{P L}]

In fact, if ν is a lift of ν_{M}^{P L} then there is a smooth structure on M_{ν} for which ν classifies the stable normal bundle. We would now like to determine the number of distinct smooth structures which the P L manifold M admits. It amounts to the same thing, but gives us a little more freedom, to determine the smoothings of M where we define a smoothing of M to be a P L-homeomorphism f : M_{ν} → M where M_{ν} is a smooth manifold. Such a smoothing defines the smooth structure f^{−1∗}M_{ν} on M. Two smoothings f_{0} : M_{ν_{0}} → M and f_{1} : M_{ν_{1}} → M are called concordant if there is a smooth structure on M × I which restricts to be f_{0}^{−1∗}M_{ν_{0}} on M × 0 and f_{1}^{−1∗}M_{ν_{1}} on M × 1. Another theorem of [HM] states that the concordance classes of smoothings of M are in bijective correspondence with [M, P L/O] where P L/O denotes the fiber of π_{O,P L}. We shall call two smoothings f_{i} : M_{ν_{i}} → M diffeomorphic if there is a diffeomorphism g : M_{ν_{0}} → M_{ν_{1}} such that f_{0} = f_{1} ∘ g. Of course one often would rather know the classification of the smoothings of M up to diffeomorphism and these are the quotient of the concordance classes of smoothings of M under the action of the P L-homeomorphisms of M. A P L-homeomorphism h : M → M acts on a smoothing M_{ν} :→ M by composition, h · f = h ∘ f. If h is P L-isotopic to the identity then it acts trivially on any smoothing. In the case where M ≅ (S^{n})^{P L} it is known that all P L-homeomorphisms which preserve orientation are P L isotopic to the identity and hence [S^{n}, P L/O] = π_{n}(P L/O) corresponds bijectively with the set of orientation preserving diffeomorphism classes of smoothings of S^{n}, otherwise known as Θ_{n} (for n ≠ 3, 4). For i ≤ 6 it is known that π_{i}(P L/O) = 0. In general the

1. SMOOTHING THEORY 125

computation of [M, P L/O] for an arbitrary manifold is a (not necessarily easy) task in homotopy theory.

The problem of classifying the P L structures on a given topological manifold was solved by Kirby and Siebenmann [KiSi] who showed that the problem again reduced to the lifting question for the classifying maps of the stable tangent bundles. Specifically every compact topological manifold M has a stable topological normal bundle which classified by a map ν_{M}^{T op} from M to the classifying space BT op. There is a universal forgetful map π_{P L,T op} : BP L → BT op and a topological manifold M admits a P L-structure if and only if the following lifting problem can be solved.

[figure: BP L; M; BT op; π_{P L,T op}; ν_{M}^{T op}]

Moreover, the homotopy fiber of π_{P L,T op}, T op/P L, has but one non-trivial homotopy group and this is ℤ/2 in dimension 3. There is thus a single obstruction to placing a P L structure on a topological manifold M which lies in H^{4}(M; ℤ/2) and it is called the Kirby-Siebenmann invariant. If M is a topological manifold which admits a P L structure then the concordance classes of P L structures on M are in bijective correspondence with [M, T op/P L] ≅ H^{3}(M; ℤ/2).

Having reviewed the fundamental results concerning the smoothing of manifolds we now consider the P L and topological classification of highly connected manifolds of dimension 7. We shall write P^{P L} and P L^{T op} for the P L and topological manifolds underlying the smooth manifold P.

Theorem 6.1. Let P_{0} and P_{1} be highly connected, closed, smooth 7-manifolds with quadratic linking families Q(P_{0}) and Q(P_{1}). The following are equivalent:

1. P_{0} is homeomorphic to P_{1},

1. SMOOTHING THEORY 126

2. P_{0} is P L-homeomorphic to P_{1},

3. P_{0} is almost diffeomorphic to P_{1},

4. Q(P_{0}) is isometric to Q(P_{1}).

Proof. Evidently 3 implies 2 and 2 implies 1. The main result of Chapter 4 was the equivalence of 3 and 4. In dimension 7 we see that [P, P L/O] ≅ H^{7}(P; π_{7}(P L/O)) ≅ Θ_{7}. It follows that all possible smooth structures on P_{0}^{P L} are obtained by taking the connected sum of P_{0} with a homotopy sphere and thus 2 implies 3. Finally we show that 1 implies 2.

Let f : P_{0} → P_{1} be a homeomorphism. Suppose, to begin with, that P_{0} and P_{1} have torsion free third homology. Then Q(P_{i}) ≅ (ℤ^{l}, 0, β_{i}) where l is the rank of H^{4}(P_{i}), 0 indicates the trivial quadratic linking function and β_{i} is the tangential invariant of P_{i}. By Novikov’s theorem on the topological invariance of the rational Pontryagin classes, f_{!}(β_{0}) = β_{1} and hence f induces an isometry of the quadratic linking families. Hence by Theorem B, P_{0} and P_{1} are almost diffeomorphic and in particular P L-homeomorphic. If P_{0} and P_{1} have finite fourth cohomology then by Corollary 5.24 the quadratic linking functions Q(P_{i}) = (T G, q_{i}, β_{i}) are determined by their Wilkens data (b(q_{i}), β_{i}) and Kervaire-Arf invariants K(q_{i}). Now the homeomorphism f preserves the linking form since b and it also preserves the tangential invariant since by [KS] β(P_{i}) = \frac{p_{1}}{2} (P_{i}) is a topological invariant. But by Lemma 5.19 K(q_{i}) = −s̄_{1}(q_{i}) = −s̄_{1}(P_{i}) and again by [KS] the invariant s̄_{1} is a topological invariant. Thus P_{0} and P_{1} have isometric quadratic linking functions and are therefore almost diffeomorphic and in particular P L-homeomorphic.

Thus if P is a highly connected 7-manifold with either finite homology or torsion free homology, then every element of H^{3}(P; π_{3}(P L/O)) is realized by a self-smoothing f : P^{P L} → P^{T op}. Finally suppose that H^{4}(P_{0}) ≅ F ⊕T G where F is a free group and T G is a finite group. We choose a splitting of P_{0} as the connected sum of manifolds

2. THE SURGERY EXACT SEQUENCE 127

P_{0} = P_{0}(F)#P_{0}((T G) where F = H^{4}(P (F)) and T G = H^{4}(P (T G)). Given any x ∈ H^{3}(P_{0}; π_{3}(P L/O)) we may write x = y ⊕ z where y ∈ H^{3}(P_{0}(F); π_{3}(P L/O)) and z ∈ H^{3}(P_{0}(T G)^{T op}; π_{3}(P L/O)). Then by the previous arguments y and z are realized by self P L smoothings f_{y} : P_{0}(T G)^{P L} → P_{0}(T G)^{T op} and f_{z} : P_{0}(F)^{P L} → P_{0}(F)^{T op}. If we now form the connected sum f_{y}#f_{z} : P_{0}^{P L} → P_{0}^{T op} then it realizes x and is manifestly a self smoothing. Thus every element of H^{3}(P_{0}; π_{3}(P L/O)) is realized by a self P L smoothing and hence every P L manifold homeomorphic to P_{0} is P Lhomeomorphic to P_{0} and in particular P_{1}^{P L} is P L-homeomorphic to P_{0}.

2. The surgery exact sequence

We present now a rapid review of the surgery exact sequence which is an important tool for the classification of manifolds. We begin with a pair of finite CW-complexes (Y, X) which we assume to be simply connected and where we allow X to be empty. Let Cat denote either of the smooth (O), piecewise linear (P L) or topological (Top) categories. We wish to know if there is a homotopy equivalence f : (W, M) → (Y, X) from a compact pair of Cat-manifolds (W, M) to (Y, X). A necessary condition for such a homotopy equivalence is of course that the pair (Y, X) satisfy Poincaré duality. That is, for some n there is a class [Y, X] ∈ H_{n}(Y, X; ℤ) such that ∩[Y, X] : H^{k}(Y, X) ≅ H_{n−k}(Y). As the pair (Y, X) is a finite complex there a large N and an embedding (Y, X) ↪ D^{N}, S^{N−1}) where (D^{N}, S^{N−1}) is N-disc with boundary the (N − 1)-sphere S^{N−1}. Inside D, the complex Y has a regular neighborhood N. The manifold N is a smooth manifold and retracts onto Y via a retraction we denote by r : N → Y. It is a consequence of Poincaré duality for the pair (Y, X) that the map r|_{∂N}, when considered as a fibration, has homotopy fiber the homotopy type of a sphere. The stable fiber homotopy type of the fibration defined by r|_{∂N} is independent of the embedding and is a homotopy invariant of the pair (Y, X) ([Sp]). It is called

2. THE SURGERY EXACT SEQUENCE 128

the Spivak normal bundle of (Y, X) and is classified by a map ν : Y → BG where BG is the classifying space for spherical fibrations. Now if Y were a Cat manifold then r would be a stable normal bundle of Y and r|_{∂N} would be the sphere bundle of the stable normal bundle. Hence a further necessary condition for (Y, X) to have the homotopy type of a compact manifold with boundary is that its Spivak normal bundle have a Cat bundle reduction. Assuming that the dimension of (Y, X) as a Poincaré space is greater than 5 then it is a theorem of Browder [Bro] and Novikov [No] (in the smooth and P L categories) and Kirby and Siebenmann [KiSi] (in the T op category)^{1} that a Cat reduction of the Spivak normal bundle of a simply connected Poincaré space is in fact a sufficient condition for it to have the homotopy type of a Cat manifold.

With an answer to the question of when a homotopy type (Y, X) contains the homotopy type of a manifold, surgery proceeds to the task of classifying Cat-manifolds homotopy equivalent to (Y, X) up to Cat-isomorphism. However, the primary objects which surgery theory seeks to classify are in fact smoothings of a finite CW -pair (Y, X). The Cat-structure set of Y, S^{Cat}(Y), consists of equivalence classes of Catsmoothings in Y where a Cat smoothing in Y is an orientation preserving homotopy equivalences of pairs f : (M, ∂M) → (Y, X) from a compact Cat-manifold with boundary to (Y, X). Two homotopy equivalences (M_{1}, f_{1}) and (M_{2}, f_{2}) are equivalent in S^{Cat}(Y) if there is a Cat-isomorphism g : M_{1} → M_{2} such that f_{1} is homotopy equivalent to f_{2} ∘ g. In order to compute the set of Cat smoothings in Y one maps S^{Cat}(Y) to the set N^{Cat}(Y) of degree one normal maps in Y modulo normal bordism.

1By their extension of transversality methods to T op [KiSi, III])

2. THE SURGERY EXACT SEQUENCE 129

We now define this set. A degree one normal map to Y is a bundle map (f, f̂)

ν_{M} →^{f̂} ν
↓  ↓
(M, ∂M) →^{f} (Y, X)

from the stable normal bundle ν_{M} of a Cat-manifold M to a Cat-bundle over Y such that map induced on homology by f carries the fundamental class of (M, ∂M) to [Y, X]. The equivalence between degree one normal maps is simply that of bordism with boundary over the bundle ν → Y. Specifically, two degree one normal maps

\begin{matrix}ν_{M_{0}} & →^{f̂_{0}} & ν_{0} \\ ↓ & & ↓ \\ (M_{0}, ∂M_{0}) & →^{f_{0}} & (Y, X) \end{matrix}  \begin{matrix}ν_{M_{1}} & →^{f̂_{1}} & ν_{1} \\ ↓ & & ↓ \\ (M_{1}, ∂M_{1}) & →^{f_{1}} & (Y, X) \end{matrix}

are normally bordant if there exists a compact manifold W and a degree one normal map (F, F̂)

ν_{W} →^{F̂} ν × I
↓  ↓
W →^{F} Y × I

with the following two properties: (i) ∂W = M_{1} ∪ M_{2} ∪ N where N is a compact manifold with ∂N = N ∩ (M_{1} ∪ M_{2}) = ∂(M_{1} ∪ M_{2}), and (ii) (F |_{M_{i}}, F̂|_{M_{i}}) = (f_{i}, f̂_{i}).

To report the essential results of simply connected surgery theory we first define the following abelian groups.

L_{n}(ℤ) = \{ \begin{matrix}Z, & if n = 4 k; \\ Z_{2}, & if n = 4 k + 2; \\ 0, & otherwise. \end{matrix} \} , L_{n}(ℤ → ℤ) = 0.

The following two facts are well-known, see [Wa5] or [MM].

Fact 6.2. (1) Let M be a simply connected manifold and let G/Cat denote the fiber of the map BCat → BG, then N^{Cat}(M) ≅ [M, G/Cat].

2. THE SURGERY EXACT SEQUENCE 130

(2) If k > 5 there is a pair of surgery exact sequences of sets

L_{k+1}(ℤ → ℤ) ⟶ S^{Cat}(W^{k}) ⟶^{η^{Cat}} N^{Cat}(W^{k}) ⟶ L_{k}(ℤ → ℤ)
↓ i^{∗} ↓ i^{∗} ↓ ↓
L_{k}(ℤ)  ⟶ S^{Cat}(∂W^{k−1}) ⟶^{η^{Cat}} N^{Cat}(∂W^{k−1}) ⟶ L_{k−1}(ℤ)

where exactness means that each set has a preferred element ∗ and that at every stage the pre-image of ∗ is the image of the previous map. The base point of both S^{Cat} and N^{Cat} is the the class of the identity map Id : W → W or Id : ∂W → ∂W. The base point of the L-groups is the identity element. In fact there is an action of the left-most L-group on S^{Cat} and two smoothings in S^{Cat} are in the same orbit if and only if they have the same image under η^{Cat}. In the smooth category this action is connect sum in the domain of a smoothing with a homotopy sphere bounding a parallelizable manifold and in either the P L or T op categories the action is trivial and η is injective. Finally, the vertical maps i^{∗} are given by restricting a smoothing or degree one normal map to its boundary.

We now illustrate this theory in the P L case for the manifolds (W, ∂W) = (L, P). In this case n = 8, L_{7}(e) = 0 and η^{P L} is a bijection. Thus both P L-structure sets and P L-normal invariant sets coincide. Moreover the normal invariants are extremely easy to calculate since [L, G/P L] and [P, G/P L] can be identified, via the the primary obstruction to null homotopy, with H^{4}(L; π_{4}(G/P L)) ≅ H^{4}(L; ℤ) and H^{4}(P; π_{4}(G/P L)) ≅ H^{4}(P; ℤ) respectively. The surgery exact sequences become the following commutative squares.

\{ \begin{matrix}S^{P L}(L) & ≅^{η} & N^{P L}(L) \\ i^{∗} ↓ & & i^{∗} ↓ \\ S^{P L}(P) & ≅^{η} & N^{P L}(P) \end{matrix} \} ≅ \{ \begin{matrix}H^{4}(L) & ≅ & H^{4j}(L) \\ i^{∗} ↓ & & i^{∗} ↓ \\ H^{4}(P) & ≅ & H^{4j}(P) \end{matrix} \}

3. CONSTRUCTING SMOOTHINGS IN L AND P 131

We emphasize that the vertical maps i^{∗} are given by simply restricting a P L-structure or P L-normal invariant to the structure or normal invariant on the boundary. Since the map i^{∗} : H^{4}(L) → H^{4}(P) is always onto it follows that every degree one normal map in P is normally bordant to the restriction of a degree one normal map in L and thus that every equivalence class of P L-smoothings in P has a representative which is the restriction of a P L-smoothing in L.

3. Constructing smoothings in L and P

Let SG(n) denote the H-space of orientation preserving self homotopy equivalences of S^{n−1}. In what follows the groups SG(4) and SO(4), and in particular π_{3}(SG(4)) and π_{3}(SO(4)), shall play a crucial role so we begin by digressing somewhat to describe these groups and the relationships between them.

We start with SO(4). Let | | denote the standard metric on ℝ^{4} and identify ℝ^{4} with the quaternions, ℍ. Then S^{3} = {x ∈ ℍ : |x| = 1} is the set of unit quaternions. For y ∈ ℍ and x ∈ S^{3} we define the function

\begin{aligned}a(m, n) : & S^{3} & → & SO(4) \\ & x & ↦ & [y ↦ x^{n+m} · y · x^{−m}] \end{aligned}

where · indicates quaternionic multiplication. The mapping a(m, n) is manifestly linear and orientation preserving and since |x^{n+m} · y · x^{−m}| = |x|^{n+m}|y||x|^{−m|} = |y| and it follows that a(m, n)(x) is an element of SO(4) for each x ∈ S^{3}. We define ᾱ(m, n) to be the homotopy class of a(m, n). It is well known that the mapping

\begin{matrix}ℤ ⊕ ℤ & → & π_{3}(SO(4)) \\ (m, n) & ↦ & ᾱ(m, n) \end{matrix}

3. CONSTRUCTING SMOOTHINGS IN L AND P 132

defines an isomorphism of groups. Given the function a(m, n) we define the associated twist A(m, n) as follows

\begin{aligned}A(m, n) : & S^{3} × D^{4} & → & S^{3} × D^{4} \\ & (x, y) & ↦ & (x, a(m, n)(y)). \end{aligned}

Identifying S^{3} ×D^{4} with ∂D^{4} ×D^{4} we may use A(m, n) to glue two copies of D^{4} ×D^{4} together thereby form a D^{4}-bundle over S^{4}, ξ(m, n) : L(m, n) → S^{4}. We define the Euler number of ᾱ(m, n) to be the Euler number of ξ(m, n), E(ᾱ(m, n)) := E(ξ(m, n)). The following facts are also well known and may be found in [Hu] (r = ᾱ(1, 0) and s = ᾱ(0, 1)).

1. The Euler number of ᾱ(m, n), E(ᾱ(m, n)), is n.

2. The stabilization of ᾱ(m, n), S(ᾱ(m, n)), is 2m + n ∈ ℤ = π_{3}(SO).

3. If i : SO(3) ↪ SO(4) denotes the standard inclusion with induced map i_{∗} : π_{3}(SO(3)) ↪ π_{3}(SO(4)), and if m ∈ Z = π_{3}(SO(3)), then i_{∗}(m) = ᾱ(m, 0).

By restricting an element of SO(n) to S^{n−1} we obtain an orientation preserving homotopy equivalence of S^{n−1} and this defines an inclusion i_{n} : SO(n) ↪ SG(n).

Lemma 6.3. There is an isomorphism π_{3}(SG(4)) ≅ ℤ/12 ⊕ ℤ such that

i_{4∗}(ᾱ(m, n)) = (m (mod 12), n).

Proof. Let SF (n) denote the sub-H-space of SG(n + 1) consisting of orientation preserving homotopy equivalences of S^{n} which fix a point p. By the adjoint correspondence there is an isomorphism A_{n,j} : π_{j}(SF (n)) ≅ π_{j+n}(S^{n}). We relate the homotopy groups of SF (n) to SG(n + 1) by noting that SF (n) is the fiber of the fibration

\begin{aligned}ev : & SG(n + F) & → & S^{n} \\ & f & ↦ & f(p). \end{aligned}

3. CONSTRUCTING SMOOTHINGS IN L AND P 133

Moreover, ev|_{SO(n+1)} : SO(n + 1) → S^{n} is the standard fibration with fiber SO(n) ⊂ SF (n). When n = 3, ev|_{SO(4)} has a section s : S^{3} → SO(4) given by s(x) = a(0, n)(x). Applying the long exact homotopy sequence to this pair of fibrations at π_{3} we obtain the following commutative diagram with exact rows.

[figure: 0; ℤ; ℤ ⊕ ℤ; ℤ; 0; 0; π_{3}(SO(3)); π_{3}(SO(4)); π_{3}(S^{3}); 0; 0; π_{3}(SF (3)); π_{3}(SG(4)); π_{3}(S^{3}); 0; 0; ℤ/12; ℤ/12 ⊕ ℤ; ℤ; 0; ≅; ≅; ≅; j_{∗}; i_{4∗}; ev|_{SO(4)∗}; Id; ≅; ≅; ev_{∗}; ≅]

To conclude the proof we note the following. Firstly, if n ∈ π_{3}(S^{3}) then s_{∗}(n) = ᾱ(0, n) ∈ π_{3}(SO(4)). Secondly, if m ∈ π_{3}(SO(3)) then i_{∗}(m) = ᾱ(m, 0). Finally, if j_{n} : SO(n) ↪ SF (n) denotes the inclusion of one fiber into the other then the homomorphism

A_{n,j} ∘ j_{n∗} : π_{j}(SO(n)) → π_{n+j}(S^{n})

is the usual J-homomorphism and the J-homomorphism J_{3,3} : π_{3}(SO(3)) → π_{6}(S^{3}) is known to be the surjection ℤ → ℤ/12 ([JW2]).

Now, let L be a 8-dimensional handlebody and let the quadratic function of L be κ(L) = κ(H, λ, α). Recall from Section 2.2 that for every choice of basis, {v_{1} … v_{l}}, for H there is a corresponding presentation for the handlebody L,

ϕ = ⊔_{i=1}^{l}ϕ_{i} : ⊔_{i=1}^{l}(S^{3} × D^{4})_{i} ↪ ∂D^{8},

and that L is diffeomorphic to the adjunction space D^{8} ∪_{ϕ} ⊔_{i=1}^{l}(D^{4} × D^{4})_{i}. The isotopy invariants of ϕ are the linking numbers c_{ij} of the attaching spheres ϕ̄(S_{i}^{3}) where S_{i}^{3} = (S^{3} × 0)_{i}, and the framing invariants ᾱ(i) ∈ π_{3}(SO(4)) which we denote

3. CONSTRUCTING SMOOTHINGS IN L AND P 134

now by ba_{i}. They are related to λ and α as follows:

c_{ij} = λ(v_{i}, v_{j}) (i ≠ j),  E(ᾱ_{i}) = λ(v_{i}, v_{i}),  S(ᾱ_{i}) = α(v_{i}).

Lemma 6.4. Suppose that the handlebody L has a presentation

ϕ = ⊔_{i=1}^{l}ϕ_{i} : ⊔_{i=1}^{n}(S^{3} × D^{4})_{i} ↪ ∂D^{8}

with invariants c_{ij}, (i ≠ j), and ᾱ_{i}, and let γ̄_{i} = ᾱ(m_{i}, n_{i}) ∈ π_{3}(SO(4)). Then

ϕ(γ̄_{1}, …, γ̄_{l}) := ⊔_{i=1}^{l}ϕ_{i} ∘ A(m_{i}, n_{i})

is a presentation with invariants c_{ij}, (i ≠ j), and ᾱ_{i} + γ̄_{i}.

Proof. As A(m_{i}, n_{i})(x, 0) = (x, 0) for every x ∈ S^{3}, ϕ_{i} ∘ A(m_{i}, n_{i})|_{S_{i}^{3}} = ϕ_{i}|_{S_{i}^{3}}. Hence the attaching spheres and linking numbers c_{ij} are unchanged. Recall that the framing invariant ᾱ_{i} is defined as the homotopy class of a(ϕ_{i}) where

ψ_{i} : ϕ_{i}((S^{3} × S^{4})_{i}) → ϕ̄_{i}(S_{i}^{3}) × D^{4}

is a framing of ϕ_{i}((S^{3} × D^{4})_{i}) which extends to a framing of the normal bundle of a 4-disc embedded in D^{8} bounding ϕ_{i}((S^{3} × 0)_{i}) and where

\begin{aligned}ψ_{i} ∘ ϕ_{i} : & (S^{3} × D^{4})_{i} & → & ϕ̄_{i}(S_{i}^{3}) × D^{4} \\ & (x, y) & ↦ & (ϕ̄_{i}(x), a(ϕ_{i})(x)(y)). \end{aligned}

After isotopy we may assume that a(ϕ_{i}) = a(p_{i}, q_{i}) for some p_{i} and q_{i}. For the presentation ϕ(γ̄_{1}, …, γ̄_{l}) we consider instead the diffeomorphism ψ_{i} ∘ ϕ_{i} ∘ A(m_{i}, n_{i}) which is

\begin{aligned}A(p_{i}, q_{i}) ∘ A(m_{i}, n_{i}) : & (S^{3} × D^{4})_{i} & → & ϕ̄_{i}(S^{3}) × D^{4} \\ & (x, y) & ↦ & (ϕ̄_{i}(x), x^{m_{i}+n_{i}+p_{i}+q_{i}}.y.x^{−m_{i}−p_{i}}). \end{aligned}

It follows that the framing functions for a(ϕ(γ̄_{1}, …, γ̄_{l})_{i}) are a(n_{i} + p_{i}, m_{i} + q_{i}) and that the framing invariants for ϕ(γ̄_{1}, …, γ̄_{l}) are

ᾱ(n_{i} + p_{i}, m_{i} + q_{i}) = ᾱ(p_{i}, q_{i}) + ᾱ(n_{i}, m_{i}) = ᾱ_{i} + γ̄_{i}.

3. CONSTRUCTING SMOOTHINGS IN L AND P 135

We shall abbreviate γ̄_{1}, …, γ̄_{l} to γ̄ and denote the handlebody obtained from the twisted presentation by L(γ̄).

Lemma 6.5. Let L, ϕ, L(γ̄) and ϕ(γ̄) be as in the previous lemma and let γ̄_{i} ∈ Ker[i_{4∗} : π_{3}(SO(4)) → π_{3}(SG(4))] for i = 1, …, l. Then there is a homotopy equivalence

g(γ̄) : (L(γ̄), ∂L(γ̄)) → (L, ∂L).

Proof. We make the identifications

L = D^{8} ∪_{ϕ} ⊔_{i=1}^{l}(D^{4} × D^{4})_{i} and L(γ̄) = D^{8} ∪_{ϕ(γ̄)} ⊔_{i=1}^{l}(D^{4} × D^{4})_{i}.

The map g(γ̄) shall take each handle of L to the corresponding handle of L(γ̄). On the zero handle, D^{8}, we define g(γ̄) to be the identity. Now consider the ith handles of L and L(γ̄) which we denote by (D^{4} × D^{4})_{i}. The map g(γ̄) has been defined only on (∂D^{4} × D^{4})_{i} where it is ϕ_{i}^{−1} ∘ ϕ(γ̄)_{i}. Now ϕ(γ̄)_{i} = ϕ_{i} ∘ A(m_{i}, n_{i}) where A(m_{i}, n_{i}) is the twisting corresponding to a(m_{i}, n_{i}). It follows that g(γ̄)|_{(∂D^{4}×D^{4})_{i}} = A(m_{i}, n_{i}). As a(m_{i}, n_{i}) represents γ̄_{i} and i_{4∗}(γ̄_{i}) = 0 ∈ π_{3}(SG(4)) we conclude from Lemma 6.3 firstly that each n_{i} = 0 and that 12 divides m_{i}. Secondly, for every j, i_{4} ∘ a(12j, 0) : S^{3} → SG(4) is null homotopic so we let b(12j, 0) : D^{4} → SG(4) be such a null homotopy and we define an extension of A(12j, 0) as follows. For x ∈ D^{4}, y ∈ S^{3}, t ∈ [0, 1] and ty ∈ D^{4} let

\begin{aligned}B(12j, 0) : & D^{4} × D^{4} & → & D^{4} × D^{4} \\ & (x, ty) & ↦ & (x, tb(12j, 0)(x)(y)). \end{aligned}

On each handle (D^{4} × D^{4})_{i} we define g(γ̄) to be B(m_{i}, 0).

We now show that g(γ̄) is a homotopy equivalence of pairs. The boundary of L is obtained by performing surgeries on ∂D^{8} described by the embeddings ϕ,

∂L = (∂D^{8} − [⊔_{i=n}^{l}ϕ_{i}((∂D^{4} × D^{4})_{i})] ∪_{ϕ|_{⊔{l \atop i=1}∂D^{4}×∂D^{4}}} ⊔_{i=1}^{l}(D^{4} × ∂D^{4})_{i}.

3. CONSTRUCTING SMOOTHINGS IN L AND P 136

A similar statement holds true for ∂L(γ̄). Since g(γ̄)|_{∂D^{8}} = Id_{∂D^{8}} and for each i, B(m_{i}, 0)((D^{4} × ∂D^{4})_{i}) = (D^{4} × ∂D^{4})_{i}, we see that g(γ̄) maps the boundary of L(γ̄) to the boundary of L. The generators of H_{4}(L) and H_{4}(L(γ̄)) may be represented by embedded spheres constructed by gluing the cores of each handle to 4-discs embedded in D^{8} and bounding the corresponding attaching spheres. Since B(m_{i}, 0) is the identity map when restricted to the core of each handle and since g(γ̄) is the identity on the zero handle we see that g(γ̄) carries the generators of H_{4}(L) to the generators of H_{4}(L(γ̄)). Since L and L(γ̄) have non-zero homology only in dimensions 0 and 4 we conclude that g(γ̄)_{∗} : H_{∗}(L(γ̄)) ≅ H_{∗}(L). Since all the groups involved are free we conclude that g(γ̄)^{∗} : H^{∗}(L) ≅ H^{∗}(L(γ̄)). We now apply the naturality of the Poincaré duality isomorphisms H^{8−∗}(L) ≅ H_{∗}(L, ∂L) and H^{8−∗}(L(γ̄)) ≅ H_{∗}(L(γ̄), ∂L(γ̄)) to conclude that g(γ̄)_{∗} : H_{∗}(L(γ̄), ∂L(γ̄)) ≅ H_{∗}(L, ∂L). Applying the five lemma to the long exact homology sequences of (L, ∂L) and (L(γ̄), ∂L(γ̄)), it follows that (g(γ̄)|_{∂L(γ̄)})_{∗} : H_{∗}(∂L(γ̄)) ≅ H_{∗}(∂L). Thus g(γ̄) induces a homology equivalence of simply connected pairs and so by Whitehead’s Theorem g(γ̄) induces a homotopy equivalence of pairs.

Corollary 6.6. Let L be a handlebody with quadratic function κ(H, λ, α) and for each ϵ ∈ 24·H^{∗} let L_{ϵ} be the handlebody with quadratic function κ(L_{ϵ}) = κ(H, λ, α+ϵ). Then there is a homotopy equivalence

g_{ϵ} : (L_{ϵ}, ∂L_{ϵ}) → (L, ∂L)

such that g_{ϵ∗} = Id : H → H.

Proof. Let {v_{1}, …, v_{l}} be a basis for H and let ϕ and ϕ_{ϵ} be presentations for L and L_{ϵ} respectively which correspond to this basis. If the invariants of ϕ are c_{ij} and ᾱ_{i} then the invariants for ϕ_{ϵ} are c_{ij} and ᾱ_{i} + γ̄_{i} where γ̄_{i} = ᾱ(ϵ(v_{i})/2, 0). As 24 divides ϵ, 12 divides ϵ(v_{i})/2 for each i and so by Lemma 6.3, γ̄_{i} ∈ Ker(i_{4∗}). Thus we

3. CONSTRUCTING SMOOTHINGS IN L AND P 137

apply Lemma 6.5 and set f_{ϵ} = g(γ̄). The induced map g_{ϵ_{∗}} is the identity because, as we noted in the proof of Lemma 6.5, g_{ϵ} maps the embedded sphere representing v_{i} in L_{ϵ} to the corresponding sphere in L. Hence for each i, g_{ϵ∗}(v_{i}) = v_{i}. Since {v_{1}, …, v_{n}} is a basis for H, g_{ϵ∗} = Id_{H}.

Lemma 6.7. Let g_{ϵ} : L(κ(H, λ, α + ϵ)) → L(κ(H, λ, α)) be the homotopy equivalence of handle bodies defined in Corollary 6.6. Then g_{ϵ} has P L-normal invariant η^{P L}(g_{ϵ}) = ϵ/24 ∈ N^{P L}(L) ≅ H^{4j}(L) = H^{∗}.

Proof. Having previously identified N^{P L}(L) with H^{4}(L; π_{4}(G/P L)) we consider the change of coefficient map to U : H^{4}(L; π_{4}(G/P L)) → H^{4}(L; π_{4}(BP L)). If [f] ∈ H^{4}(L; π_{4}(G/P L)) corresponds to a degree one normal map

ν_{M}  →^{f̂}  ν
↓  ↓
(M, ∂M) →^{f} (L, P),

then U([f]) corresponds to the difference of bundles ν_{L} − ν ∈ [L, BP L]. Now the map π_{4}(G/P L) → π_{4}(BP L) is ·24 : ℤ → ℤ as it has cokernel π_{3}^{S} ≅ ℤ/24 and π_{4}(BP L) ≅ π_{4}(BO) ≅ ℤ. Thus U is injective and all normal invariants in L are detected by the bundle data. We wish to calculate U(η(g_{ϵ})). Now, as g_{ϵ} is a homotopy equivalence η(g_{ϵ}) is the degree on normal map

ν_{L_{ϵ}}  →^{ĝ_{ϵ}} (g_{ϵ}^{−1})^{∗}ν_{L_{ϵ}}
↓  ↓
(L_{ϵ}, ∂L_{ϵ}) →^{g_{ϵ}} (L, ∂L)

3. CONSTRUCTING SMOOTHINGS IN L AND P 138

where L = L(κ(λ, α)) and L_{ϵ} = L(κ(λ, α + ϵ)). Note that ν_{L} = −α is the inverse of the stable tangent bundle of L and that g_{ϵ} induces the identity on cohomology.

U(η(g_{ϵ})) = ν_{L} − (g_{ϵ}^{−1})^{∗}ν_{L_{ϵ}}
= (−α) − (g_{ϵ}^{−1})^{∗}(−α − ϵ)
= ϵ

Since U is the coefficient homomorphism which is multiplication by 24, η(g_{ϵ}) = ϵ/24.

Corollary 6.8. Let L be an 8-dimensional handlebody with quadratic function κ = κ(H, λ, α) and boundary P = ∂L. If g_{ϵ} : L_{ϵ} := L_{(}λ, α + ϵ) → L(λ, α) is the homotopy equivalence of Corollary 6.6 and let f_{ϵ} = ∂L_{ϵ}, then,

1. S^{P L}(L) = {[g_{ϵ} : L_{ϵ} → L]| ϵ ∈ 24 · H^{∗}},

2. S^{P L}(P) = {[f_{ϵ} : ∂L_{ϵ} → ∂L]| ϵ ∈ 24 · H^{∗}}.

Proof. The first assertions follows immediately from Lemma 6.7 and the isomorphism S^{P L}(L) ≅ H^{4}(L) which was discussion in Section 2 following Fact 6.2. The second assertion follows from the fact that the map i^{∗} : S^{P L}(L) → S^{P L}(P) is onto, which was demonstrated in the same discussion.

It is a simple task to use Corollary 6.8 to descend from the P L classification of handlebodies L to their homotopy classification. We introduce some extra notation so that the theorem we state is of the same form as Wall’s corresponding result for closed manifolds ([Wa1] Lemma 8). We let J : π_{3}(SO) → π_{3}^{S} denote the stable Jhomomorphism which is known to be the surjection ℤ → ℤ/24. The J-homomorphism induces a coefficient homomorphism for any space X, J_{∗} : H^{∗}(X; π_{3}(SO(4)) → H^{∗}(X; π_{3}^{s}).

Theorem 6.9. Two handlebodies L(κ_{0}) and L(κ_{1}) with quadratic functions κ_{0} = κ(H_{0}, λ_{0}, α_{0}) and κ_{1} = κ(H_{1}, λ_{1}, α_{1}) are homotopy equivalent if and only if there is

3. CONSTRUCTING SMOOTHINGS IN L AND P 139

an isometry of their intersection forms, Θ : H_{0} → H_{1}, λ_{1} ∘ (Θ × Θ) = λ_{0}, such that J_{∗}(α_{1}) = J_{∗}(Θ^{∗}(α_{0})). Moreover, every such isomorphism Θ is induced by a homotopy equivalence f : L_{0} → L_{1}.

Proof. From Corollary 6.8 we deduce that the handlebody L_{1} is homotopic to L_{0} if and only if L_{1} is P L-homeomorphic to L(κ(H_{0}, λ_{0}, α_{0} + ϵ)) for some ϵ ∈ 24 · H^{∗}. Now by the classification of handlebodies (see Remark 2.6) this is the case if and only if there is an isometry

Θ : κ(H_{1}, λ_{1}, α_{1}) ≅ κ(H_{0}, λ_{0}, α_{0} + ϵ))

for some ϵ ∈ 24 · H^{∗}. But by Proposition 2.10 there is such an isometry if and only if there is an isomorphism Θ : H_{1} → H_{0} such that λ_{1}∘(Θ×Θ) = λ_{0} and Θ^{∗}(α_{0}+ϵ) = α_{1} for some ϵ ∈ 24 · H^{∗}. But Ker(J_{∗}) = 24 · H^{∗} and thus there is an ϵ ∈ 24 · H^{∗} such that Θ^{∗}(α_{0} + ϵ) = α_{1} if and only if J_{∗}(Θ^{∗}(α_{0})) = J_{∗}(α_{1}).

Finally, suppose Θ : H_{1} → H_{0} is as in the statement of this theorem. Then by the remarks above there is an ϵ ∈ 24 · H_{0}^{∗} such that Θ defines an isometry κ_{1} ≅ κ(H_{0}, λ_{0}, α_{0} + ϵ). This isometry is implemented by a diffeomorphism g which we compose with g_{ϵ} from Corollary 6.6. Since g_{ϵ∗} = Id, (g_{ϵ} ∘ g)_{∗} = g_{∗} = Θ.

In order to apply Corollary 6.8 to the homotopy classification of highly connected 7-manifolds we first remind the reader of the definition of the characteristic quadratic linking family induced by a characteristic quadratic function and define the related concept of a J-quadratic linking family. Suppose that κ(H, λ, α) is a (characteristic) quadratic function. Recall that G and F denote respectively Cok(λ̂) and Ker(λ̂) ant that τ_{G} : G → G^{∗∗} and ρ : H → H/F denote the canonical maps whose sets of sections are Sec(ρ) and Sec(τ_{G}). Each section Ψ ∈ Sec(ρ) induces a splitting H^{∗} = F^{∗}(Ψ) ⊕ H^{∗}(Ψ) and every x ∈ H^{∗} splits as x = (p_{F} (Ψ)^{∗}(x|_{F}), x(Ψ)) where p_{F} (Ψ) : H → F. In addition, each Ψ defines a section Φ(Ψ) in Sec(τ_{G}) and also a

3. CONSTRUCTING SMOOTHINGS IN L AND P 140

nondegenerate quadratic function κ(Ψ). The quadratic linking function of κ, δ^{c}(κ), is the triple (G, q^{c}(κ), β) where β = [α] is the image of α in G and q^{c}(κ) : Sec(τ) → Q(b(λ)) is the function defined by q^{c}(Φ(Ψ)) = q^{c}(κ(Ψ)). Here Q(b) denotes the set of quadratic linking functions which refine the linking form induced by λ, b(λ). A crucial property of q^{c}(κ) is that it is equivariant with respect to the actions of Hom(G^{∗∗}, T G) on Sec(τ_{G}) and Q(b(λ)) which are defined, for ϕ ∈ Hom(G^{∗∗}, T G), Φ ∈ Sec(τ_{G}) and q ∈ Q(b(λ)) by ϕ · Φ = Φ − ϕ and ϕ · q = q_{ϕ(τ_{G}(β)/2)}. We now make the following definitions for an abelian group G containing the even element β and with torsion subgroup T G. Firstly, given a linking form b on T G, let Q_{J}(b) be the subset of the power set of Q(b) consisting of sets of the form

S(q) := {q_{12a}|a ∈ T G, q ∈ Q(b)}.

We let Hom(G^{∗∗}, T G) act on Q_{J}(b) by setting ϕ · S(q) := S(q_{ϕ(τ(β))/2}) where ϕ ∈ Sec(τ_{G}) and S(q) ∈ Q_{J}(b).

Definition 6.10. 1. A J-quadratic linking family is a triple (G, q_{J}^{c}, β_{J}) where G is a finitely generated abelian group, β_{J} is an even element of G ⊗ ℤ/24 and q_{J}^{c} : Sec(τ_{G}) → Q_{J}(b) is a function for some linking form b on T G and which satisfies the following properties. (A) q_{J}^{c} is equivariant with respect to the actions of Hom(G^{∗∗}, T G) on Sec(τ_{G}). (B) If q_{J}^{c}(Φ) = S(q) then β_{J} = (Φ(τ(β)) + β(q)) ⊗ 1 ∈ G ⊗ ℤ/24.

2. For any characteristic quadratic linking family, Q = (G, q^{c}, β), the of J-quadratic linking family of Q is the triple Q_{J} = (G, q_{J}^{c}, β_{J}), where β_{J} = β ⊗ 1 ∈ G ⊗ ℤ/24 and

q_{J}^{c}(Φ) := S(q^{c}(Φ)) = {q^{c}(Φ)_{12a}| a ∈ T G}.

3. Let P be a highly connected 7-dimensional manifold. The J-quadratic linking family of P, Q_{J}(P), is the J-quadratic linking family of Q(P).

3. CONSTRUCTING SMOOTHINGS IN L AND P 141

Recall from Section 2.10 that an isomorphism of abelian groups θ : G_{0} → G_{1} induces a bijection τ_{θ} : Sec(τ_{G_{0}}) → Sec(τ_{G_{1}}). If Q_{J0} = (G_{0}, q_{J0}^{c}, β_{J0}) and Q_{J1} = (G_{1}, q_{J1}^{c}, β_{J1}) are J-quadratic linking families then an isomorphism θ : G_{0} ≅ G_{1} shall be called a J-isometry if (θ ⊗ Id)(β_{J0}) = β_{J1} and if for every Φ ∈ Sec(τ_{G_{0}}),

q_{J1}^{c}(τ(θ(Φ)) ∘ θ|_{T G_{0}} = q_{J0}^{c}(Φ)

where, if q_{J1}^{c}(τ(θ(Φ)) = S(q), then q_{J1}^{c}(τ(θ(Φ)) ∘ θ|_{T G_{0}} = S(q ∘ θ|_{T G_{0}}).

Theorem 6.11. Let P_{0} and P_{1} be a pair of 2-connected 7-manifolds with Jquadratic linking families Q_{J}(P_{0}) = (G_{0}, q_{J0}^{c}, β_{J0}) and Q_{J}(P_{1}) = (G_{1}, q_{J1}^{c}, β_{J1}). Then there is a homotopy equivalence f : P_{0} → P_{1} such that f_{!} = θ : H^{4}(P_{0}) ≅ H^{4}(P_{1}) if and only if θ defines a J-isometry of J-quadratic linking families from Q_{J}(P_{0}) to Q_{J}(P_{1}).

Proof. In dimension 7 every highly connected manifold bounds a handlebody. So for i = 0, 1, let P_{i} = ∂L_{i} and let the handlebody L_{i} have quadratic function κ_{i} = κ(H_{i}, λ_{i}, α_{i}). Suppose that f : P_{0} → P_{1} is a homotopy equivalence. As there are no obstructions to smoothing 7-dimensional P L-manifolds, S^{P L}(P_{1}) ≅ S^{O}(P_{1})/Θ_{7}, where Θ_{7} acts by connected sum with a homotopy sphere in the domain of a homotopy equivalence. Applying Corollary 6.8 we conclude that for some ϵ ∈ 24 · H_{1}^{∗} there is an almost diffeomorphism f′ : P_{0} → ∂L_{ϵ} such that f = f_{ϵ} ∘ f′, where g_{ϵ} : L_{ϵ} := L(κ(H_{1}, λ_{1}, α_{1} + ϵ)) → L_{1} is the homotopy equivalence of Lemma 6.6 and f_{ϵ} = g_{ϵ}|_{L_{ϵ}}. By Theorem B, f′_{!} defines an isometry of quadratic linking families f′_{!} : Q^{c}(P_{0}) ≅ δ^{c}(κ(λ_{1}, α_{1} + ϵ)). It is clear from the definitions that an isometry of quadratic linking families induces an isometry of the induced J-quadratic linking families. Moreover, f_{!} = f_{ϵ!} ∘ f′_{!} = f′_{!} since f_{ϵ!} = Id_{G_{1}}. So it remains to check that Id_{G_{1}} defines a Jisometry from Q_{J}(∂(L_{ϵ})) to Q_{J}(∂L_{1}) = Q_{J}(P_{1}). Clearly since 24 divides ϵ, β_{1} ⊗ 1 = (β_{1} + [ϵ]) ⊗ 1 ∈ G_{1} ⊗ ℤ/24. For any Φ ∈ Sec(τ_{G_{1}}), let Ψ ∈ Sec(ρ_{1}) be such that

3. CONSTRUCTING SMOOTHINGS IN L AND P 142

Φ(Ψ) = Φ. Recalling that ϵ(Ψ) denotes the component of ϵ in H_{1}^{∗}(Ψ) we calculate that

q_{J}^{c}(κ(λ_{1}, α_{1} + ϵ))(Φ) = S[q^{c}(κ(λ_{1}, α_{1} + ϵ))(Ψ)]
= S[q^{c}(κ(λ_{1}(Ψ), α_{1}(Ψ) + ϵ(Ψ))]
= S[q^{c}(κ(λ_{1}(Ψ), α_{1}(Ψ))_{[ϵ(Ψ)/2]}] by Lemma 2.37
= S[q^{c}(κ(λ_{1}(Ψ), α_{1}(Ψ))]  since 12 divides [ϵ(Ψ)/2]
= S[q^{c}(κ_{1})(Ψ)]
= S[q^{c}(κ_{1})(Φ)]
= q_{J1}^{c}(Φ).

Hence Id_{G_{1}} defines a J-isometry from Q_{J}(L_{ϵ}) to Q_{J}(P_{1}).

Conversely, suppose that θ is a J-isometry from Q_{J}^{c}(P_{0}) to Q_{J}^{c}(P_{1}). Let Φ_{0} ∈ Sec(τ_{G_{0}}) be any section and choose Ψ_{0} ∈ Sec(ρ_{0}) and Ψ_{1} ∈ Sec(ρ_{1}) such that Φ_{0} = Φ(Ψ_{0}) and τ_{θ}(Φ_{0}) = Φ(Ψ_{1}). Then for some a ∈ T G_{1},

q^{c}(κ_{1}(Ψ_{1}))_{12a} ∘ θ|_{T G_{0}} = q^{c}(κ_{1})(τ_{θ}(Φ_{0}))_{12a} ∘ θ|_{T G_{0}} = q^{c}(κ_{0})(Φ_{0}).

Applying Lemma 2.37 in reverse, choose an ϵ′ ∈ H_{1}^{∗}(Ψ_{1}) which is divisible by 24 and so that q^{c}(κ(λ_{1}(Ψ_{1}), α_{1}(Ψ_{1}) + ϵ′)) = q^{c}(κ_{1}(Ψ_{1}))_{12a}. It follows that

q^{c}(κ(λ_{1}(Ψ_{1}), α_{1}(Ψ_{1}) + ϵ′)) ∘ θ|_{T G_{0}} = q^{c}(κ_{0})(Φ_{0})

and in particular that θ(β(q^{c}(Φ_{0})) = [α(Ψ_{1}) + ϵ′]. Now β_{0} = Φ_{0}(τ_{G_{0}}(β_{0})) + β(q^{c}(Φ_{0})) and so θ(β_{0})−[α(Ψ_{1})+ϵ′] = θ(Φ_{0}(τ_{G_{0}})(β_{0})) = τ_{θ}(Φ_{0})(β_{1}). Now α_{1} −α_{1}(Ψ_{1}) ∈ F_{1}^{∗}(Ψ_{1}) and so we may choose ϵ′′ ∈ F_{1}^{∗}(Ψ_{1}) such that θ(β_{0}) = [α_{1}+ϵ′+ϵ′′] and ϵ′′ is necessarily divisible by 24 as (θ(β_{0}) ⊗ 1) = β_{1} ⊗ 1 ∈ G_{1} ⊗ ℤ/24. Setting ϵ = ϵ′ + ϵ′′ ∈ 24 · H_{1}^{∗}, we see that (α_{1} + ϵ)(Ψ_{1}) = α_{1}(Ψ_{1}) + ϵ′ and we calculate that

q^{c}(κ(λ_{1}, α_{1} + ϵ))(τ_{θ}(Φ_{0})) ∘ θ|_{T G_{0}} = q^{c}(κ(λ_{1}, α_{1} + ϵ))(Ψ_{1}) ∘ θ|_{T G_{0}}
= q^{c}(κ(λ_{1}(Ψ_{1}), α_{1}(Ψ_{1}) + ϵ′)) ∘ θ|_{T G_{0}}
= q^{c}(κ_{0})(Φ_{0}).

4. SMOOTH S^{3} BUNDLES OVER S^{4} 143

By the equivariance property of characteristic quadratic linking families, θ defines an isometry from Q^{c}(P_{0}) to δ^{c}(κ(λ_{1}, α_{1} + ϵ)). By Theorem B, there is an almost diffeomorphism f′ : P_{0} → ∂L(κ(λ_{1}, α_{1} + ϵ)) such that f′_{!} = θ. Composing f′ with f_{ϵ} we obtain a homotopy equivalence f_{ϵ}∘f′ such that (f_{ϵ}∘f′)_{!} = f_{ϵ!}∘f′_{!} = Id_{G_{1}}∘θ = θ.

4. Smooth S^{3} bundles over S^{4}

As we remarked in the introduction, the recent interest in 2-connected 7-manifolds was generated by Grove and Ziller’s geometric result that the total space of every S^{3}- bundle over S^{4} carries a metric of non-negative sectional curvature [GZ]. These manifolds have a lengthy history in differential topology which is covered in [CE] where the manifolds are classified up to diffeomorphism, P L-homeomorphism, homeomorphism and homotopy. The proofs in [CE] used the results of Wilkens [Wi1, Wi2] and calculations of the invariants s_{1} and s̄_{1}. By Hatcher’s result on the homotopy type of Diff(S^{3}) [Ha] the classification holds equally well for smooth bundles over S^{4} with fiber a differentiable 3-sphere. We now present the classification of these total spaces using the results and notation of this thesis.

Recall from Section 3 the oriented ℝ^{4}-bundles over S^{4}, ξ(m, n). Let L_{m,n} be the handle body which is the total space of the disc bundle of ξ(m, n) and let P_{m,n} be the boundary of L_{m,n}. We orient L_{m,n} and P_{m,n} by fixing an orientation on S^{4} and using the orientation of the vector bundle on the fiber.^{2} There is a diagram of commuting

2We warn the reader that this orientation convention differs from the one use in [CE]. There, following James and Whitehead, we fixed an orientation on S^{4} and oriented the fiber so that the Euler number of ξ(m, n) was positive. The two conventions agree for n ≥ 0. The later one seems more natural.

4. SMOOTH S^{3} BUNDLES OVER S^{4} 144

fiber bundles

[figure: D^{4}; L_{m,n}; S^{4}; S^{3}; P_{m,n}; S^{4}.; π_{L}; i; π_{P}; Id]

Let s_{4} and s^{4} denote the preferred generators of H_{4}(S^{4}) and H^{4}(S^{4}) respectively. The group H_{4}(L_{m,n}) is infinite cyclic with a preferred generator s such that π_{L∗}(s) = s_{4}. The bundle ξ(m, n) has Euler number n and stable invariant n + 2m, thus s · s = n and the tangential invariant of L_{m,n}, α_{m,n}, satisfies α_{m,n}(s) = (n + 2m).^{3} The group H^{4}(P_{m,n}) is isomorphic to ℤ/n with preferred generator e = π_{P}^{∗} (s^{4}). By applying Proposition 1.4 and Remark 2.6 it is easy to see that L_{m_{0},0} (resp. P_{m_{0},0}) is diffeomorphic to L_{m_{1},0} (resp. P_{m_{1},0}) if and only if m_{0} = ±m_{1}. We assume now that n ≠ 0. The algebraic invariants of L_{m,n} and P_{m,n} are as follows (where j and k are integers, we have used Proposition 2.57 to compute s_{1}(P_{m,n}) and s̄_{1}(P_{m,n}) and ϵ(n) = n/|n|):

\begin{matrix}α_{m,n} : ℤ → ℤ & j · s = (n + 2m)j, \\ λ_{m,n} = λ_{n} : ℤ × ℤ ⟶ ℤ & (j · s, k · s) ↦ njk, \\ κ_{m,n} : ℤ ⟶ ℤ & j · s ↦ nj^{2} + (n + 2m)j, \\ β_{m,n} ∈ ℤ/n & β_{m,n} = 2m · e \\ b_{m,n} = b_{n} : ℤ/n × ℤ/n ⟶ ℚ/ℤ & (j · e, k · e) ↦ \frac{1}{n}(jk) mod ℤ \\ q_{m,n} : ℤ/n ⟶ ℚ/ℤ & je ↦ \frac{1}{2n}(j^{2} + 2mj) mod ℤ \\ s_{1}(P_{m,n}) = \frac{1}{28} · \frac{1}{8n}((n + 2m)^{2} − nϵ(n)) & s_{1}(P_{m,n}) = \frac{1}{8n}((n + 2m)^{2} − nϵ(n)). \end{matrix}

When n ≠ 0, P_{m,n} is a rational homology sphere which falls under the classification Theorem A. There (1 and 5) it is stated that the isometry class of q_{m,n} determines the topological classification and (2) that the isometry class of q_{m,n} along with s_{1}(P_{m,n}) determines the smooth classification. Moreover by (4), the quadratic function q_{m,n} is

3The cohomology class α_{m,n} ∈ H^{4}(L_{m,n}) is also equal to the Spin characteristic class, \frac{p_{1}}{2} (L_{m,n}).

4. SMOOTH S^{3} BUNDLES OVER S^{4} 145

determined by the pair (b(n), β_{m,n}) and s̄_{1}(P_{m,n}) ∈ ℚ/ℤ. Lastly, by Theorem A (6), P_{m_{0},n_{0}} is homotopy equivalent to P_{m_{1},n_{1}} if and only if q_{m_{0},n_{0}} ≅ (q_{m_{1},n_{1}})_{12je} for some integer j. It is then a matter of definitions and some simple algebra that the following classification holds.

Theorem 6.12 ([CE]). Let n ≠ 0 and ϵ = ±1. The total spaces P_{m_{0},n} and P_{m_{1},ϵn} are (1.) diffeomorphic, (2.) homeomorphic or (3.) homotopic if and only if the following (appropriately numbered) condition holds:

1. (a) 4m_{0}(n + m_{0}) + (n^{2} − n) ≡ ±[4m_{1}(n_{1} + ϵm_{1}) + ϵ(n^{2} − n)] (mod 224n), (b) 2m_{0} ≡ 2αm_{1} (mod n) for some α ∈ ℤ/n such that α^{2} ≡ ±ϵ (mod n).

2. (a) 4m_{0}(n + m_{0}) + (n^{2} − n) ≡ ±[4m_{1}(n_{1} + ϵm_{1}) + ϵ(n^{2} − n)] (mod 8 n), (b) 2m_{0} ≡ 2αm_{1} (mod n) for some α ∈ ℤ/n such that α^{2} ≡ ±ϵ (mod n).

3. (a) m_{0} ≡ αm_{1} (mod(n, 12)) where α^{2} ≡ ±ϵ (mod(n, 12)).

The diffeomorphism/homeomorphism/homotopy equivalence is orientation preserving or reversing as the relevant equations hold with either a + sign or − sign.

There are more detailed arithmetical calculations in [CE] concerning how many of the smooth structures within a given topological type are realized by S^{3}-bundles over S^{4} and of the numbers α ∈ ℤ/n such that α^{2} = ±1 (mod n). We conclude by noting that the manifolds P_{1,8} and P_{5,8} are tangentially homotopy equivalent but not homeomorphic as we now show. One computes that s̄_{1}(P_{1,8}) = 9/16 ≠ 1/16 = s̄_{1}(P_{5,8}) and hence P_{1,8} and P_{5,8} are not homeomorphic by Theorem A 3. The quadratic functions P_{1,8} and P_{5,8} are respectively q^{3}(1)_{5e} and q^{3}(1)_{e}. Since, q^{3}(1)_{5e+12e} = q^{3}(1)_{e}, it follows by Theorem A 6 that P_{1,8} and P_{5,8} are homotopy equivalent via a homotopy equivalence f which induces the identity on H^{4}(P_{1,8}) = ℤ/8 = H^{4}(P_{5,8}). Since

4. SMOOTH S^{3} BUNDLES OVER S^{4} 146

β(P_{1,8}) = β(P_{5,8}) = 2 ∈ ℤ/8, it follows that f is tangential.^{4} This pair of manifolds yields a counter example to Theorems C and Theorem 5.10 of [MTW] where it is claimed, amongst other things, that all tangentially equivalent 2-connected 7manifolds are homeomorphic.

4Note that the same conclusion holds for every pair of manifolds (P^{7}, ∗P^{7}) for which (b(P), β(P)) ≅ (b(∗P), β(∗P)) but s̄_{1}(P) ≠ s̄_{1}(∗P). There is such a pair for every finite abelian group with order divisible by 8.

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