MULTIPLICITIES OF DUPIN HYPERSURFACES

Stephan Stolz

University of Notre Dame and Max–Planck–Institut für Mathematik, Bonn

March, 1997

Abstract. All known examples of isoparametric hypersurfaces of spheres are of the following two types:

(1) The homogeneous examples, which are principal orbits of isotropy representations of symmetric spaces of rank 2.

(2) The Clifford examples of Ferus–Karcher–Münzner, which are constructed

from modules over Clifford algebras.

Our main result, which extends work of Münzner, Thorbergsson and Grove–Halperin, is that any isoparametric hypersurface has the same number of principal curvatures and the same multiplicities as some hypersurface of type (1) or (2). More generally, this holds for any Dupin hypersurface. The proof is homotopy theoretic; the main tools are the Hopf invariant and the EHP–sequence.

1. Introduction

From a geometric point of view the simplest hypersurfaces of spheres are the isoparametric hypersurfaces. We recall that a hypersurface M^{n} ⊂ S^{n+1} is called isoparametric if its principal curvatures (the eigenvalues of the shape operator) are constant. A more general class of hypersurfaces are the Dupin hypersurfaces for which the number of distinct eigenvalues of the shape operator at x ∈ M is required to be independent of x and the i-th eigenvalue λ_{i}(x) is constant on the leaves of the foliation defined by the eigenspaces corresponding to λ_{i}(x).

Elie Cartan classified those isoparametric hypersurfaces whose number g of distinct principal curvatures is equal to 1, 2, or 3 [Ca] (all of them are homogeneous). Moreover, it was shown that all Dupin hypersurfaces with g = 1, 2, 3 are Lie equivalent to isoparametric surfaces (this is classical for g = 1, due to Cecil and Ryan [CR] for g = 2, and to Miyaoka [Miy] for g = 3).

For g > 3 a classification of isoparametric hypersurfaces or Dupin hypersurfaces is not known. However we have a lot of information about the number g and the possible multiplicities m_{i} (m_{i} is the dimension of the eigenspace corresponding to the eigenvalue λ_{i} for i = 1, 2, …, g; here we assume that the eigenvalues are ordered according to size: λ_{1} < λ_{2} < ⋯ < λ_{g}). Münzner has shown for isoparametric

Typeset by AMS-TEX

1991 Mathematics Subject Classification. primary: 53C40; secondary: 53C42, 55Q25, 55P99.

Key words and phrases. isoparametric hypersurfaces, Dupin hypersurfaces, vector fields on spheres, Hopf invariant, EHP–sequence.

Partially supported by NSF Grant DMS–9504418 and the Max–Planck–Institut in Bonn.

Hypersurfaces that g = 1, 2, 3, 4, or 6 and that m_{i+2} = m_{i}, i ∈ ℤ/g [Mü1, Satz 1(b)]; in particular, all multiplicities m_{i} are determined by m_{1} and m_{2}; moreover, m_{1} = m_{2} for g odd. These statements continue to hold for Dupin hypersurfaces by work of Thorbergsson [Th, Remark on p. 497] and Grove-Halperin [GH, §2]. For g = 6, Grove and Halperin proved that the only possible multiplicities are m_{1} = m_{2} = 1 or m_{1} = m_{2} = 2 [GH, Thm. C(iv)]; they also show: if g = 4 and m_{1} = m_{2}, then m_{1} = 1 or 2; if g = 4, and 2 ≤ m_{1} < m_{2}, then m = m_{1} + m_{2} is odd. Stronger restrictions for isoparametric surfaces have been obtained by Abresch [Ab], and very recently, by Fang [Fa].

We note that the shape operator A of the embedding M ⊂ S^{n+1} involves the choice of a normal direction; making the opposite choice replaces A by −A and hence switches m_{1} and m_{2}. Hence we may assume m_{1} ≤ m_{2} without loss of generality. The main result of this paper is the following theorem.

1.1 Theorem. Let M^{n} ⊂ S^{n+1} be a Dupin hypersurface with g = 4 distinct principal curvatures and multiplicities m_{1} ≤ m_{2}. Then (m_{1}, m_{2}) = (2, 2) or (4, 5), or m_{1} + m_{2} + 1 is a multiple of 2^{ϕ(m_{1}−1)}. Here ϕ(l) is the number of integers s with 1 ≤ s ≤ l and s ≡ 0, 1, 2, 4 mod 8.

The above restrictions on the multiplicities are sharp. To show this we recall that the known examples of isoparametric hypersurfaces are of the following two types.

(1) Hsiang and Lawson noticed that every homogeneous isoparametric hypersurface occurs as a principal orbit of the isotropy representation of some symmetric space of rank 2 [HL]. These were studied in detail by Takagi and Takahashi [TT]. They show in particular that for g = 6 the multiplicities (1, 1) and (2, 2) occur, and for g = 4 among others the multiplicities (2, 2) and (4, 5).

(2) Extending work of Ozeki-Takeuchi [OT], Ferus-Karcher-Münzner [FKM] associate to each module V over the Clifford algebra C_{l} an isoparametric hypersurface with g = 4, m_{1} = l+1, and m_{2} = dim V −m_{1}−1 (cf. [KFM, 3.3 and Satz 4.1]). We note that the dimension of the irreducible modules over the Clifford algebra C_{l} is 2^{ϕ(l)} (there are two irreducible modules for l ≡ 0 mod 4, and one otherwise). It follows that positive integers (m_{1}, m_{2}) with m_{1} ≤ m_{2} occur as multiplicities of isoparametric hypersurfaces of Clifford type if and only if m_{1} + m_{2} + 1 is a multiple of 2^{ϕ(m_{1}−1)}.

Summarizing the above discussion we conclude:

1.2 Corollary. For any Dupin hypersurface M^{n} ⊂ S^{n+1} there is a homogeneous isoparametric hypersurface or an isoparametric hypersurface of Clifford type with the same multiplicities.

This result possibly suggests that every isoparametric hypersurface in a sphere is equivalent to a homogeneous isoparametric hypersurface or an isoparametric hypersurface of Clifford type. However we should mention that there are Dupin hypersurfaces not Lie equivalent to any isoparametric hypersurface; examples were provided by Pinkall and Thorbergsson [PT] and Miyaoka and Ozawa [MO].

In the next section we will outline the proof of Theorem 1.1; a description of the contents of the subsequent sections can be found at the end of section 2.

I would like to thank Hans-Joachim Baues, Linus Kramer and Matthias Kreck for helpful discussions and especially Gudlaugur Thorbergsson for introducing me to the subject of Dupin hypersurfaces and for sharing his insights.

§2. Outline of the proof of Theorem 1.1

Münzner showed that an isoparametric hypersurface M^{n} ⊂ S^{n+1} leads to a decomposition of the sphere

(2.1)  S^{n+1} = Dγ_{1} ∪_{M} Dγ_{2},

as the union of two linear disc bundles Dγ_{i} ⟶ M_{i} over smooth manifolds M_{i} with common boundary ∂Dγ_{1} = ∂Dγ_{2} = M [Mü1, Satz 4(d)], where the fibers of the fiber bundle p_{i}: M = ∂Dγ_{i} ⟶ M_{i} are spheres of dimension m_{i}. Geometrically the manifolds M_{i} (considered as submanifolds of S^{n+1} ⊂ M via the identification (2.1)) are focal manifolds of M, and γ_{i} is the normal bundle of M_{i}. The above statements continue to be true for Dupin hypersurfaces by work of Thorbergsson [Th] and Grove-Halperin [GH, §2].

Let M be a Dupin hypersurface of dimension n with g distinct principal curvatures and multiplicities m_{1}, …, m_{g}. Then n = m_{1} + ⋯ + m_{g}, and since as mentioned in the introduction m_{i+2} = m_{i} for i ∈ ℤ/g, we have

(2.2)  n = \frac{gm}{2} for m = m_{1} + m_{2}.

The results of Münzner, Thorbergsson and Grove-Halperin mentioned in the introduction concerning the number of principal curvatures of isoparametric (resp. Dupin) hypersurfaces and their multiplicities are consequences of this decomposition result, and the following purely topological result.

2.3 Theorem. Assume that S^{n+1} decomposes in the form (2.1), and let g be given by equation (2.2). Then

(1) (Münzner [Mü1, Satz A]) g = 1, 2, 3, 4, or 6.

(2) (Grove-Halperin [GH, Thm. C(iv)(d)]) If g = 6, then m_{1} = m_{2} = 1 or m_{1} = m_{2} = 2.

(3) (Grove-Halperin [GH, Thm. C(iv)(c)]) If g = 4 and m_{1} = m_{2}, then m_{1} = 1 or 2; if g = 4, and 2 ≤ m_{1} < m_{2}, then m = m_{1} + m_{2} is odd.

Similarly, our main result Theorem 1.1 is a consequence of the decomposition result and the following theorem.

2.4 Theorem. Assume that S^{n+1} decomposes in the form (2.1), and let g be given by equation (2.2). Assume that g = 4 and m_{1} ≤ m_{2}. Then (m_{1}, m_{2}) = (2, 2) or (4, 5), or m_{1} + m_{2} + 1 is a multiple of 2^{ϕ(m_{1}−1)}.

We would like to draw attention to the similarities between this theorem and Adams’ result concerning vector fields on spheres. In terms of the numerical function ϕ(l), Adams’ result can be phrased as follows [Ad, Thm. 1.1] (cf. [Hu, Prop. 11.2]).

2.5 Theorem (Adams). If there are l linearly independent vector fields on the sphere S^{m} then m + 1 is a multiple of 2^{ϕ(l)}.

We recall that the classical Hurwitz-Radon-Eckmann construction produces l linearly independent vector fields on the sphere S^{m} starting with a module V over the Clifford algebra C_{l} with dim V = m + 1 (cf. [Hu, Chap. 11, Thm. 2.2, Thm. 2.4 and 2.5]). Adams’ result shows that the maximal possible number of linearly independent vector fields is achieved by this construction. Similarly, Theorem 2.4 shows that all possible multiplicities of Dupin hypersurfaces except (m_{1}, m_{2}) = (2, 2), (4, 5) are realized by the Clifford examples of Ferus-Karcher-Münzner. In fact, the Hurwitz-Radon-Eckmann construction and the Ferus-Karcher-Münzner construction are closely related: if β is the subbundle of the tangent bundle T S^{m} of vectors orthogonal to the l vector fields produced by the Hurwitz-Radon-Eckmann construction, then its sphere bundle S(β) is diffeomorphic to a focal manifold of the corresponding isoparametric hypersurface of Clifford type (cf. [FKM, Satz 4.2(ii)]).

We will prove Theorem 2.4 by reducing it to a homotopy theoretic result of Adams which is the technical core of his theorem on vector fields. To state this result we need some notation. Let H_{l} be the real Hopf line bundle over the real projective space ℝℙ^{l} of dimension l, and let T (kH_{l}) be the Thom space of the Whitney sum of k copies of H_{l} (we recall that the Thom space of a vector bundle α is obtained from the disc bundle of α by collapsing the sphere bundle to a point). We remark that the standard CW-structure on ℝℙ^{l} (with one cell in every dimension s for 0 ≤ s ≤ l) gives T (kH_{l}) the structure of a CW-complex with one cell in every dimension s for k ≤ s ≤ k + l.

2.6 Theorem (Adams). Assume that the top cell of T ((m − l)H_{l}) splits off stably, i.e., there is a stable map s: S^{m} ⟶ T ((m − l)H_{l}) such that the composition

S^{m} ⟶^{s} T ((m − l)H_{l}) ⟶^{p} S^{m}

has mapping degree one, where p is the projection map which collapses the (m − 1)-skeleton of T ((m − l)H_{l}) to a point. Then m + 1 is a multiple of 2^{ϕ(l)}.

We recall that a stable map from a pointed space X to a pointed space Y is a map f: Σ^{k}X ⟶ Σ^{k}Y between suitable supensions of X and Y. To facilitate comparison with the literature we want to mention that results like Theorem 2.6 above are frequently phrased in terms of statements about the stunted projective space ℝℙ_{m−l}^{m} = ℝℙ^{m}/ℝℙ^{m−l−1} (obtained from ℝℙ^{m} by collapsing the subspace ℝℙ^{m−l−1} ⊂ ℝℙ^{m} to a point) which is homeomorphic to T ((m − l)H_{l}) (cf. [Hu, Chap. 15, Thm. 1.8]).

Roughly speaking, Theorem 2.6 is the Spanier-Whitehead dual of Theorem 1.2 in [Ad]. For the convenience of the reader we will make this relationship precise at the end of this section.

It is a classical result of James that the existence of l linearly independent vector fields on S^{m} implies that the top cell of T ((m − l)H_{l}) splits off stably (cf. [Hu, Chap. 15, Thm. 9.4]). It shows that Adams’ technical result Theorem 2.6 implies his result on vector fields on spheres (Theorem 2.5).

One way of proving James’ result proceeds in the following two steps.

(1) If S^{m} admits l linearly independent vector fields, then the Thom space T (τS^{m}) of the tangent bundle of S^{m} desuspends l times (i.e., T (τS^{m}) is homotopy equivalent to Σ^{l}X for some space X).

(2) If T (τS^{m}) desuspends l times then the top cell of T ((m − l)H_{l}) splits off stably.

We remark that step 1 is rather obvious (cf. [Hu, Chap. 15, Cor. 1.6]). The following propositions are analogous to steps (1) resp. (2) above.

2.7 Proposition. Assume that S^{n+1} decomposes in the form (2.1) with 2 ≤ m_{1} < m_{2}, n = 2m, m = m_{1} + m_{2}. If (m_{1}, m_{2}) ≠ (2, 2), (4, 5), then (M_{1} ∧ M_{2})^{(2m)} (the 2m-skeleton of M_{1} ∧ M_{2}) desuspends l times for some l with ϕ(l) ≥ ϕ(m_{1} − 1).

2.8 Proposition. Assume that S^{n+1} decomposes in the form (2.1) with 2 ≤ m_{1} < m_{2}, n = 2m, m = m_{1} + m_{2}. If (M_{1} ∧ M_{2})^{(2m)} desuspends at least l times, 0 < l ≤ m_{1} − 1, then the top cell of T ((m − l)H_{l}) splits off stably.

Together with Adams’ technical result (2.6) these propositions imply Theorem 2.4. We note that we may assume for the proof of Theorem 2.4 that m_{1} ≥ 2, since otherwise the conclusion that m + 1 is a multiple of 2^{ϕ(m_{1}−1)} is trivially satisfied. We can also assume m_{1} < m_{2}, since the case m_{1} = m_{2} was settled by Grove-Halperin (cf. Thm. 2.3(3)). We observe that the condition n = 2m, m = m_{1} + m_{2} in Propositions 2.7 and 2.8 is equivalent to g = 4 by equation (2.2).

For the proof of Proposition 2.7 we will need the following result which is a corollary of Proposition 2.8 and Adams’ result (2.6).

2.9 Corollary. If (M_{1} ∧ M_{2})^{(2m)} desuspends at least l times, 0 < l ≤ m_{1} − 1, then m + 1 is a multiple of 2^{ϕ(l)}.

Comments on the Proof of Theorem 2.6. We recall that a closed manifold M is Spanier-Whitehead dual to the Thom space T (−τ), where τ is the tangent bundle of M, and we write −τ for any bundle which is an inverse to τ in the sense that its Whitney sum with τ is trivial (e.g., we could take −τ to be the normal bundle of some embedding of M into Euclidean space). More generally, the Thom space T (α) of some vector bundle over M is Spanier-Whitehead dual to T (−(α ⊕ τ)) (cf. [Hu, Chap. 15, Thm. 3.7]). In particular, if M is the real projective space ℝℙ^{l} of dimension l, then τ is stably (i.e., after adding trivial bundles) isomorphic to (l + 1)H, and hence T ((m − l)H_{l}) is Spanier-Whitehead dual to T (−(m + 1)H_{l}).

The assumption that the top cell of T ((m − l)H_{l}) splits off stably implies that in the Spanier-Whitehead dual space T (−(m + 1)H_{l}) the bottom cell splits off stably (the space T (−(m + 1)H_{l}) is called s-coreducible in the terminology of [Ad], [Hu]). This implies that the vector bundle −(m + 1)H_{l} represents the trivial element in the group J(ℝℙ^{l}) consisting of stable J-equivalence classes of vector bundles over ℝℙ^{l} (cf. [Hu, Chap. 15, Thm. 7.7]). We recall that two vector bundles α, β are stably J-equivalent if and only if the sphere bundles S(α ⊕ ϵ^{r}) and S(β ⊕ ϵ^{s}) are fiber homotopy equivalent for some r, s, where ϵ^{r} denotes the trivial vector bundle of dimension r.

Adams showed that J(ℝℙ^{l}) is the cyclic group of order 2^{ϕ(l)} generated by H_{l}. In particular, if −(m + 1)H_{l} represents the trivial element of J(ℝℙ^{l}), then m + 1 is a multiple of 2^{ϕ(l)}. □

The paper is organized as follows. Sections 3 and 4 are devoted to preliminaries; we recall Münzner’s results concerning the cohomology of the manifolds M, M_{1} and M_{2} in section 3, while in section 4 we discuss the EHP-sequence. This is our main homotopy theoretic tool. We prove Proposition 2.8 in section 5. Section 6 contains the proof of Proposition 2.7 under the additional assumption (m_{1}, m_{2}) ≠ (8, 9), (8, 11). In section 7 we show that the cases (m_{1}, m_{2}) = (8, 9), (8, 11) can’t occur.

3. The cohomology of M, M_{1}, and M_{2}

The goal of this section is to recall some results of Münzner concerning the cohomology of the pieces M, M_{1}, and M_{2} of a decomposition (2.1) of the sphere S^{n+1}. Following his conventions we write H^{∗}( ) for the cohomology with coefficients in R, where R = ℤ if M_{1} and M_{2} are both orientable, and R = ℤ/2 otherwise.

3.1 Lemma. If m_{1} and m_{2} are both larger than one, then M, M_{1} and M_{2} are simply connected; in particular, they are orientable.

Proof. Apply van Kampen’s Theorem to the decomposition (2.1). □

Applying the Mayer-Vietoris sequence to the decomposition (2.1) we conclude that

(3.2)  H^{q}(M_{1}) ⊕ H^{q}(M_{2}) ⟶^{p_{1}^{∗}⊕p_{2}^{∗}} H^{q}(M)

is an isomorphism for 0 < q < n. This shows in particular, that the homomorphisms p_{1}^{∗} and p_{2}^{∗} are injective; we will use these maps to identify H^{∗}(M_{i}) with a subring of H^{∗}(M). The cohomology of the manifolds M_{i} is (additively) given as follows.

3.3 Proposition ([Mü2, Satz 5]). Assume that S^{n+1} decomposes in the form (2.1), and let m = m_{1} + m_{2}. Then the cohomology groups of M_{i}, i = 1, 2, are given by

H^{q}(M_{i}) = \{ \begin{matrix}R & q ≡ 0, m_{i+1} mod m, 0 ≤ q < n \\ 0 & otherwise \end{matrix}

This implies in particular that dim H^{∗}(M_{i}) = \frac{1}{2} dim H^{∗}(M) = g, where g is defined by equation (2.2), i.e., n = gm/2.

Next we discuss some of the multiplicative structure of H^{∗}(M). Let 1 ∈ H^{0}(M) the unit of the cohomology ring H^{∗}(M). For g ≥ 2, let a_{i} ∈ H^{m_{i+1}}(M_{i}) ≅ R be a generator. The Leray–Hirsch Theorem applied to the fiber bundle M ⟶ M_{i} with fiber S^{m_{i}} leads to the following results.

3.4 Proposition ([Mü2, Satz 6]). For g ≥ 2 the cohomology ring H^{∗}(M) is a free module over H^{∗}(M_{i}) with basis {1, a_{i+1}}, i = 1, 2.

3.5 Corollary. For g ≥ 3 there is a choice of generators b_{i} ∈ H^{m}(M_{i}) ≅ R such that a_{1}a_{2} = b_{1} + b_{2}.

Proof. The elements b_{1}, b_{2} form a basis of H^{m}(M) due to the isomorphism (3.2). Hence a_{1}a_{2} = λ_{1}b_{1} + λ_{2}b_{2} with λ_{i} ∈ R. Proposition 3.3 implies that also the elements b_{1}, a_{1}a_{2} form a basis and hence λ_{2} = ±1. By the same argument λ_{1} = ±1, which proves the lemma. □

4. Hopf invariant and EHP-sequence

In this section we recall some well-known constructions from homotopy theory. In particular, we describe the (generalized) EHP-sequence, the main tool in the proof of Propositions 2.7 and 2.8. We include a discussion of how the EHP-sequence follows from the configuration space model for Ω^{k}Σ^{k}X, since we will need the geometric formulation of the EHP-sequence (Corollary 4.6, used in the proof of Lemma 7.2) and compatibilty of the Hopf invariant with suspensions (Lemma 4.7, used in the proof of Proposition 2.8). Also, this discussion might help to make the EHP-sequence less mysterious for non-experts.

4.1 Suspension homomorphism. Let X, Y be pointed spaces (i.e., X, Y are topological spaces with distinguished points x_{0} ∈ X, y_{0} ∈ Y). For technical reasons we will work in the category of compactly generated spaces with non-degenerate base points (cf. [Wh, Ch. I, §§4–6]). We recall that the smash product X ∧ Y is the pointed space obtained from the Cartesian product X × Y by collapsing the subspace X × {y_{0}} ∪ {x_{0}} × Y to a point (this is the distinguished point in X ∧ Y). The smash product S^{k} ∧ X is often denoted Σ^{k}X and is called the k-fold (reduced) suspension of X. If f: X ⟶ Y is a base point preserving map, then we get a (base point preserving) map 1∧f: S^{k} ∧X ⟶ S^{k} ∧Y, which is called the k-fold suspension of f and denoted Σ^{k}f. This construction gives in particular a homomorphism

(4.2)  Σ^{k}: π_{q}(X) ⟶ π_{q+k}(Σ^{k}X).

We recall that Freundenthal’s Suspension Theorem says that if the space X is (l − 1)-connected (i.e., if π_{i}(X) = 0 for i < l), then the suspension homomorphism (4.2) is surjective for q < 2l, and an isomorphism for q < 2l−1 (the range q < 2l−1 is called the stable range). The EHP-sequence described below is a vast generalization of Freundenthal’s Theorem. It describes in the larger range q < 3l − 2 (called the metastable range) the kernel and cokernel of the k-fold suspension map by embedding it into a long exact sequence.

The direct limit of the groups π_{q+k}(Σ^{k}X) for k → ∞ is called the q-th stable homotopy group of X and is denoted by π_{q}^{s}(X). The homomorphism (4.2) induces a homomorphism Σ^{∞}: π_{q}(X) ⟶ π_{q}^{s}(X); by Freudenthal’s Theorem, it is an isomorphism for q < 2l − 1. Following the usual custom, we write π_{q}^{s} for π_{q}^{s}(S^{0}).

For a pointed space X, let Ω^{k}X be the space of base point preserving maps S^{k} ⟶ X; we furnish it with a modification of the compact-open topology (cf. [Wh, p. 20]) and take the constant map as the base point in Ω^{k}X. If Y is a pointed space, then there is a natural homeomorphism between (base point preserving) maps Y ⟶ Ω^{k}X and maps Σ^{k}Y ⟶ X [Wh, Ch. III, (2.9)]. In particular, π_{q+k}(Σ^{k}X) can be identified with π_{q}(Ω^{k}Σ^{k}X); using this inclusion, the suspension homomorphism (4.2) can be identified with the map induced on π_{q} by the inclusion map

(4.3)  i: X ⟶ Ω^{k}Σ^{k}X

(for x ∈ X the map i(x): S^{k} ⟶ S^{k} ∧ X sends v to v ∧ x).

The following theorem is a corollary of results of May, Milgram and Segal concerning combinatorial models for Ω^{k}Σ^{k}X.

4.4 Theorem. Let X be an (l − 1)-connected space, l ≥ 1. Then the inclusion (4.3) can be factored in the form

X ⊂ F_{2}C_{k}(X) ⟶^{α_{k}} Ω^{k}Σ^{k}X,

where α_{k} is a (3l−1)-equivalence, and (F_{2}C_{k}(X), X) is an NDR-pair (cf. [Wh, Ch. I, §5]). Moreover, the quotient F_{2}C_{k}(X)/ X is homotopy equivalent to

D_{k,2}(X) =^{def} (S_{+}^{k−1} ∧ X ∧ X)/ ℤ/2.

Here S_{+}^{k−1} is the sphere S^{k−1} equipped with a disjoint base point; ℤ/2 acts on S_{+}^{k−1} by the antipodal map and on X ∧ X by permuting factors.

Before showing how this result follows from the approximation results for Ω^{k}Σ^{k}X (cf. 4.13 below), we derive some corollaries. Theorem 4.4 shows that in the metastable range q < 3l − 2 we can replace Ω^{k}Σ^{k}X by its approximation F_{2}C_{k}(X), and the inclusion i: X ⟶ Ω^{k}Σ^{k}X by the inclusion X ⟶ F_{2}C_{k}(X). Abusing notation, we will hence also write i for the latter. We will denote by p the composition of the projection map F_{2}C_{k}(X) ⟶ F_{2}C_{k}(X)/ X with the homotopy equivalence F_{2}C_{k}(X)/ X ∼ D_{k,2}(X).

We note that for an (l − 1)-connected space X the space D_{k,2}(X) is (2l − 1)-con-nected. (If X is a pointed CW-complex with only cells of dimension ≥ l, then the usual CW structure on X ∧ X has cells of dimension ≥ 2l. We note that we can think of D_{k,2}(X) as the quotient of the total space of the fiber bundle

(S^{k−1} × X ∧ X)/ ℤ/2 ⟶ S^{k−1}/ ℤ/2 = ℝℙ^{k−1}

with fiber X ∧X modulo the canonical section given by the base points of the fibers. It follows that D_{k,2}(X) has a CW structure with cells of dimension ≥ 2l.)

We recall the following result, which is a special case of the Blakers-Massey Theorem (cf.[Wh, Chap. VII, Thm. 7.12]). Let (Y, A) be a (k − 1)-connected NDR-pair, and suppose that A is (l − 1)-connected, k ≥ 2, l ≥ 2. Then the projection map (Y, A) ⟶ (Y/A, ∗) induces an isomorphism on π_{q} for q < k + l − 1 and an epimorphism for q = k + l − 1. Applying this inductively to (Y, A) = (F_{2}C_{k}(X), X), the fact that F_{2}C_{k}(X)/ X ∼ D_{k,2}(X) is (2l − 1)-connected implies that (F_{2}C_{k}(X), X) is (2l − 1)-connected. Applying the result again shows that the projection map

(F_{2}C_{k}(X), X) ⟶ (F_{2}C_{k}(X)/ X, ∗)

is a (3l − 1)-equivalence.

Now consider the long exact homotopy sequence of the pair (F_{2}C_{k}(X), X); identifying for q < 3l − 1 the groups π_{q}(F_{2}C_{k}(X)) with π_{q}(Ω^{k}Σ^{k}X) = π_{q+k}(Σ^{k}X), and π_{q}(F_{2}C_{k}(X), X)) with π_{q}(D_{k,2}(X)) leads to the following exact sequence.

4.5 Corollary (Generalized EHP-sequence). If X is an (l − 1)-connected space, then for q < 3l − 1 there is an exact sequence

π_{q}(X) ⟶^{Σ^{k}} π_{q+k}(Σ^{k}X) ⟶^{H^{k}} π_{q}(D_{k,2}(X)) ⟶^{∂} π_{q−1}(X) ⟶^{Σ^{k}} π_{q+k−1}(Σ^{k}X) ….

For k = 1, this is the classical EHP-sequence due to James (cf. [Wh, Chap. XII, Thm. 2.2]). The homomorphism H^{k} is called the (generalized) Hopf invariant. The letters EHP stand for “Einhängung” (the German word for suspension), “Hopf invariant” and “product”, respectively (if X is a sphere and k = 1, the boundary map ∂ can be interpreted as a Whitehead product).

We note that the inclusion map i: X ⟶ F_{2}C_{k}(X) lifts to the homotopy fiber of the map p: F_{2}C_{k}(X) ⟶ D_{k,2}(X), and that this lift is a (3l − 2)-equivalence (this follows by comparing the long exact homotopy sequence of the pair (F_{2}C_{k}(X), X) with the long exact sequence of the homotopy fibration). Hence we conclude:

4.6 Corollary. If X is an (l − 1)-connected space, then

X ⟶^{i} F_{2}C_{k}(X) ⟶^{p} D_{k,2}(X)

is a homotopy fibration in the metastable range (i.e., in dimensions q < 3l − 2); the corresponding exact sequence of homotopy groups can be identified with the EHPsequence.

For our proof of Proposition 2.8 it is crucial that the Hopf invariant is compatible with suspensions in the following sense.

4.7 Lemma. If X is an (l − 1)-connected space, then in the metastable range q < 3l − 2 the following diagram commutes

π_{q+k}(Σ^{k}X) ⟶^{H^{k}} π_{q}(D_{k,2}(X)) ⟶^{Σ^{r}} π_{q+r}(Σ^{r}D_{k,2}(X))
Σ^{r} ↓  ↓^{Δ_{∗}}
π_{q+k+r}(Σ^{k+r}X)  ⟶^{H^{k}}  π_{q+r}(D_{k,2}(Σ^{r}X))

Here the lower horizontal map is the Hopf invariant for the space Σ^{r}X, and the map Δ: Σ^{r}D_{k,2}(X) ⟶ D_{k,2}(Σ^{r}X) sends v ∧ [e ∧ x ∧ y] to [e ∧ (v ∧ x) ∧ (v ∧ y)] for v ∈ S^{r}, e ∈ S_{+}^{k−1}, x, y ∈ X.

We defer the proof of this lemma to the end of the section.

For calculations with the EHP-sequence, it is of course important to identify the space D_{k,2}(X). We will now discuss the well-known description of D_{k,2}(S^{n}) in terms of other spaces. It is easy to see that D_{k,2}(S^{n}) = (S_{+}^{k−1} ∧ S^{n} ∧ S^{n})/ ℤ/2 is homeomorphic to the Thom space of the vector bundle

(4.8)  (S^{k−1} × (ℝ^{n} ⊕ ℝ^{n}))/ ℤ/2

over S^{k−1}/ ℤ/2 = ℝℙ^{k−1}. Here ℤ/2 acts on ℝ^{n} ⊕ ℝ^{n} by permuting the summands. As a representation of ℤ/2, this is isomorphic to the direct sum of n copies of the non-trivial one dimensional representation ℝ^{−} and n copies of the trivial representation. Hence the vector bundle (4.8) is isomorphic to nH_{k−1} ⊕ n\underline{ℝ}, where H_{k−1} = (S^{k−1} × ℝ^{−})/ ℤ/2 is the Hopf line bundle over ℝℙ^{k−1}, and \underline{ℝ} is the one dimensional trivial real line bundle. It follows that the Thom space of the vector bundle (4.8) is homeomorphic to T (nH_{k−1}⊕n\underline{ℝ}), which in turn is homeomorphic to Σ^{n}T (nH_{k−1}) (cf. [Hu, Chap. 13, Cor. 1.6]). Moreover, the Thom space T (nH_{k−1}) is homeomorphic to the stunted projective space ℝℙ_{n}^{n+k−1} = ℝℙ^{n+k−1}/ ℝℙ^{n−1} (cf. [Hu, Chap. 13, Example 1.7]). For later reference we summarize our discussion in the following lemma.

4.9 Lemma. There are homeomorphisms

D_{k,2}(S^{n}) ≈ Σ^{n}T (nH_{k−1}) ≈ Σ^{n}ℝℙ_{n}^{n+k−1}.

Remark 4.10. Results (4.5)–(4.9) are statements for a fixed k. It is not hard to see that they are compatible for different k in the sense that there are commutative diagrams relating the results for k and k + 1. Passing to the direct limit as k approaches infinity, we get a statement for “k = ∞”. E.g., Corollary 4.5 leads to the exact sequence

π_{q}(X) ⟶^{Σ^{∞}} π_{q}^{s}(X) ⟶^{H} π_{q}(D_{2}(X)) ⟶^{∂} π_{q−1}(X) ⟶^{Σ^{∞}} π_{q−1}^{s}(X) …,

where D_{2}(X) = (S_{+}^{∞} ∧ X ∧ X)/ ℤ/2 is the direct limit of the spaces D_{k,2}(X). Corollary 4.6 leads to a homotopy fibration

X ⟶^{i} F_{2}C(X) ⟶^{p} D_{2}(X),

where F_{2}C(X) is the direct limit of the spaces F_{2}C_{k}(X). Lemma 4.9 leads to homeomorphisms

D_{2}(S^{n}) ≈ Σ^{n}T (nH) ≈ Σ^{n}ℝℙ_{n},

where H is the Hopf line bundle over ℝℙ^{∞}, and ℝℙ_{n} is the stunted projective space ℝℙ_{n} = ℝℙ^{∞}/ℝℙ^{n−1}.

Theorem 4.4 is a consequence of the existence of ‘combinatorial models’ for Ω^{k}Σ^{k}X, which we want to describe now.

4.11 Configuration space models for Ω^{k}Σ^{k}X. May, Milgram and Segal have developed combinatorial “models” for Ω^{k}Σ^{k}X (cf. [Ma], [Mil], [Se]); here we want to describe the configuration space model of May and Segal. Let C̃_{k,s} ⊂ ℝ^{k} × ⋯ × ℝ^{k} be the subspace of consisting of those tuples (v_{1}, …, v_{s}), v_{i} ∈ ℝ^{k}, for which the distance between any pair of points is ≥ 2. Given a pointed space X, define

C_{k}(X) = (∐_{s≥1} C̃_{k,s} × X × ⋯ × X)/ ∼ .

Here the equivalence relation ∼ is generated by declaring (v_{1}, …, v_{s}, x_{1}, …, x_{s}) equivalent to (v_{σ(1)}, …, v_{σ(s)}, x_{σ(1)}, …, x_{σ(s)}) for any permutation σ; moreover, if x_{i} ∈ X is the base point of X, then (v_{1}, …, v_{s}, x_{1}, …, x_{s}) is equivalent to (v_{1}, …, v̂_{i}, …, v_{s}, x_{1}, …, x̂_{i}, …, x_{s}) (obtained by deleting v_{i} and x_{i}). A point [v_{1}, …, v_{s}, x_{1}, …, x_{s}] ∈ C_{k}(X) can be thought of as a “finite subset of ℝ^{k} labelled by X” (think of x_{i} ∈ X as a label for the point v_{i} ∈ ℝ^{k}).

It is more customary to allow in the definition of C̃_{k,s} all tuples (v_{1}, …, v_{s}) for which the v_{i}’s are pairwise distinct. These two definitions of C_{n}(X) give homotopy equivalent spaces (cf. [Se, §3]). The advantage of the above definition is that it is easy to define a map

(4.12)  α_{k}: C_{k}(X) ⟶ Ω^{k}Σ^{k}X

as follows (cf. [Se, §3]). A point [v_{1}, …, v_{s}, x_{1}, …, x_{s}] ∈ C_{k}(X) determines a map ϕ: ℝ^{∞} ∪ ∞ ⟶ Σ^{k}X = D^{k}/ ∂D^{k} ∧ X defined by

ϕ(v) = \{ \begin{matrix}T_{v_{i}}^{−1}(v) ∧ x_{i} ∈ D^{k}/ ∂D^{k} ∧ X & if v ∈ B_{1}(v_{i}) \\ ∗ & otherwise \end{matrix} .

Here ∗ is the base point of D^{k}/ ∂D^{k} ∧ X, and T_{v_{i}}: D^{k} ⟶ B_{1}(v_{i}) is translation by v_{i} (B_{1}(v_{i}) denotes the ball of radius 1 around v_{i}). We note that ϕ is well-defined, since the interiors of the balls B_{1}(v_{i}) are disjoint due to the assumption that the distance between any pair of points v_{i}, v_{j} is ≥ 2. Identifying ℝ^{k} ∪ ∞ with S^{k} the above construction defines the map (4.12). According to results of May α_{k} is a weak homotopy equivalence for path connected X (i.e., it induces an isomorphism on homotopy groups, cf. [Se, Thm. 3]).

4.13 Proof of Theorem 4.4. Let F_{s}C_{k}(X) be the subspace of C_{k}(X) represented by tuples (v_{1}, …, v_{r}, x_{1}, …, x_{r}) with r ≤ s. Consider the cofibration sequence

F_{1}C_{k}(X) ⟶^{i} F_{2}C_{k}(X) ⟶^{p} F_{2}C_{k}(X)/ F_{1}C_{k}(X).

We remark:

(1) F_{1}C_{k}(X) is homotopy equivalent to X.

(2) F_{s}C_{k}(X)/ F_{s−1}C_{k}(X) is homeomorphic to ((C̃_{k,s})_{+} ∧ X ∧ ⋯ ∧ X)/ Σ_{s},

where the symmetric group Σ_{s} acts on (C̃_{k,s})_{+} and X ∧ ⋯ ∧ X by permutations. We note that this quotient is (sl − 1)-connected (cf. discussion of connectivity of D_{k,2}(X) following Theorem 4.4), and hence the composition F_{2}C_{k}(X) ⊂ C_{k}(X) ⟶^{α_{k}} Ω^{k}Σ^{k}X is a (3l − 1)-equivalence.

(3) C̃_{k,2} is ℤ/2-equivariantly homotopy equivalent to S^{k−1} with the antipodal action, and hence (2) implies that F_{2}C_{k}(X)/ X ∼ F_{2}C_{k}(X)/ F_{1}C_{k}(X) is homotopy equivalent to D_{k,2}X.

This implies Theorem 4.4. □

Proof of Lemma 4.6. The definition of the map α_{k}: C_{k}(X) ⟶ Ω^{k}Σ^{k}X (cf. 4.12) shows that the following diagram commutes.

Σ^{r}C_{k}(X) ⟶^{Σ^{r}α_{k}} Σ^{r}Ω^{k}Σ^{k}X ^{Δ}↓  ↓^{Δ} C_{k}(Σ^{r}X) ⟶^{α_{k}} Ω^{k}Σ^{k}(Σ^{r}X).

Here the vertical map Δ on the left sends v ∧ [v_{1}, … v_{s}, x_{1}, …, x_{s}] ∈ Σ^{r}C_{k}(X) to [v_{1}, … v_{s}, (v ∧ x_{1}), …, (v ∧ x_{s})] ∈ C_{k}(Σ^{r}X). The vertical map Δ on the right sends v ∧ f ∈ Σ^{r}Ω^{k}Σ^{k}X to the element of Ω^{k}Σ^{k}(Σ^{r}X) given by the map S^{k} ⟶ Σ^{k}(Σ^{r}X) = S^{r} ∧ Σ^{k}X which maps w ∈ S^{k} to v ∧ f(w).

The map Δ on the left respects the filtration of the configuration spaces; in particular, it restricts to a map Δ: Σ^{r}F_{2}C_{k}(X) ⟶ F_{2}C_{k}(Σ^{r}X). It is compatible with the map Δ defined in Lemma 4.7; i.e., the following diagram is commutative, where as in Corollary 4.6, p is the map defined after Theorem 4.4:

Σ^{r}F_{2}C_{k}(X) ⟶^{Σ^{r}p} Σ^{r}D_{k,2}(X)
Δ↓  ↓^{Δ}
F_{2}C_{k}(Σ^{r}X) ⟶^{p} D_{k,2}(Σ^{r}X)

Combining the two diagrams above and replacing the maps Δ by their adjoints Δ′, we obtain the following commutative diagram:

Ω^{k}Σ^{k}X ⟵^{α_{k}} F_{2}C_{k}(X) ⟶^{p} D_{k,2}(X)
Δ′↓  _{Δ′}↓  ↓_{Δ′}
Ω^{k+r}Σ^{k+r}X ⟵^{Ω^{r}α_{k}} Ω^{r}F_{2}C_{k}(Σ^{r}X) ⟶^{Ω^{r}p} Ω^{r}D_{k,2}(Σ^{r}X)

Now we consider the functor π_{q} (the q-th homotopy group) applied to the diagram above for q < 3l − 2. We note that

(1) The maps α_{k} and Ω^{r}α_{k} induce isomorphisms.

(2) The homomorphisms p_{∗} ∘ (α_{k})_{∗}^{−1} (resp. Ω^{r}p_{∗} ∘ (Ω^{r}α_{k})_{∗}^{−1}) coincide with the Hopf invariant H^{k} for the space X (resp. Σ^{r}X).

(3) The homomorphism induced by the left Δ′ can be identified with the suspension homomorphism Σ^{r}: π_{q+k}(Σ^{k}X) ⟶ π_{q+k+r}(Σ^{k+r}X).

(4) The homomorphism induced by the right Δ′ can be identified with the

composition π_{q}(D_{k,2}(X)) ⟶^{Σ^{r}} π_{q+r}(Σ^{r}D_{k,2}(X)) ⟶^{Δ_{∗}} π_{q+r}(D_{k,2}(Σ^{r}X).

This proves Lemma 4.6. □

5. Proof of Proposition 2.8

The main tool for the proof of Proposition 2.8 is the Hopf invariant described in the last section.

5.1 Definition of the Thom map. Let M^{n} be a framed manifold, i.e., a manifold equipped with an embedding of M into some sphere S^{n+k} and a trivialization of the normal bundle of this embedding. This allows us to identify a tubular neighbourhood T of M with D^{k} × M and the quotient space T/∂T with (D^{k} × M)/ (∂D^{k} × M) = D^{k}/∂D^{k} ∧ M_{+} = S^{k} ∧ M_{+}. The Thom map is the map

c: S^{n+k} ⟶ T/∂T = S^{k} ∧ M_{+},

which is the identity restricted to T ⊂ S^{n+k} and sends points outside of T to the base point. We will regard c as a stable map c: S^{n} ⟶ M_{+}.

Throughout the rest of this section we will assume given a decomposition of S^{n+1} of the form (2.1) with 2 ≤ m_{1} < m_{2}, n = 2m, m = m_{1} + m_{2}.

5.2 Proposition. Let c be the Thom map of the hypersurface M^{2m} ⊂ S^{2m+1}. Then the Hopf invariant of the composition

S^{2m} ⟶^{c} M^{2m} ⟶^{p_{1}∧p_{2}} (M_{1} ∧ M_{2})^{(2m)} ⊂ M_{1} ∧ M_{2}

is non-trivial.

Before proving Proposition 5.2 we want to use this result to deduce Proposition 2.8.

Proof of Proposition 2.8. By assumption there is a space X such that Σ^{l}X is homotopy equivalent to (M_{1} ∧ M_{2})^{(2m)}. Consider the following diagram which is commutative by Lemma 4.7 (applied for k = ∞; cf. remark 4.10).

π_{2m−l}^{s}(X) ⟶^{H} π_{2m−l}(D_{2}(X)) ⟶^{Σ^{l}} π_{2m}(Σ^{l}D_{2}(X))
Σ^{l}↓  ↓^{Δ_{∗}}
π_{2m}^{s}(Σ^{k+r}X) ⟶^{H} π_{2m}(D_{2}(Σ^{l}X))

Let f ∈ π_{2m−l}^{s}(X) be the element whose l-fold suspension Σ^{l}f ∈ π_{2m}^{s}(Σ^{l}X) = π_{2m}^{s}((M_{1} ∧ M_{2})^{(2m)}) is the composition (p_{1} ∧ p_{2}) ∘ c of Proposition 5.2.

Claim 1. There is a map i: S^{m−l} ⟶ X (‘inclusion of the bottom cell’) such that Σ^{l}D_{2}(i)_{∗}: π_{2m}(Σ^{l}D_{2}(S^{m−l})) ⟶ π_{2m}(Σ^{l}D_{2}(X)) is surjective.

We recall from Lemma 4.9 and Remark 4.10 that D_{2}(S^{m−l}) = D_{∞,2}(S^{m−l}) is homeomorphic to the suspended Thom space Σ^{m−l}T ((m − l)H) (where H is the Hopf line bundle over ℝℙ^{∞}). In particular, we can identify π_{2m}(Σ^{l}D_{2}(S^{m−l})) with π_{2m}(Σ^{m}T ((m − l)H)) ≅ π_{m}^{s}(T ((m − l)H)).

Let

s′: S^{m} ⟶ T ((m − l)H)

be a stable map with Δ_{∗}(Σ^{l}D_{2}(i)_{∗}(s′)) = H(f). The Hopf invariant of f is nontrivial by Proposition 5.2; since the bottom cell of D_{2}(Σ^{l}X) is in dimension 2m, this implies that H(f) induces a non-trivial map in integral homology. We conclude that hence also s′ induces a non-trivial map in H_{m}( ; ℤ).

We observe that that T ((m − l)H_{l}) (H_{l} = Hopf bundle over ℝℙ^{l}) is the m-ske-leton of T ((m − l)H), and hence up to homotopy s′ factors through a stable map s: S^{m} ⟶ T ((m − l)H_{l}).

Claim 2. For a suitable choice of s: S^{m} ⟶ T ((m − l)H_{l}), the composition with the projection map p: T ((m − l)H_{l}) ⟶ S^{m} onto the top cell has mapping degree one.

To finish the proof of Proposition 2.8, it now remains to prove claims 1 and 2. For the proof of claim 1 we recall from section 3 that M_{i} is simply connected and that H^{q}(M_{i}; ℤ) = ℤ for q = 0, m_{i+1}, m, m + m_{i+1} and trivial for all other q. Then obstruction theory shows that M_{i} is homotopy equivalent to a CW complex with one cell of dimension m_{i+1} and other cells of dimension ≥ m; it follows that we have the following homotopy equivalences

(M_{1} ∧ M_{2})^{(2m)} ∼ S^{m} ∪ cells of dimension ≥ m + m_{1}
X ∼ S^{m−l} ∪ cells of dimension ≥ m + m_{1} − l
X ∧ X ∼ S^{m−l} ∧ S^{m−l} ∪ cells of dimension ≥ 2m + m_{1} − 2l
D_{2}(X) ∼ D_{2}(S^{m−l}) ∪ cells of dimension ≥ 2m + m_{1} − 2l

Here the last line follows from the previous one by the remarks concerning a CW structure for D_{2}(X) following Theorem 4.4. The last line implies that the map

π_{2m}(Σ^{l}D_{2}(S^{m−l})) ⟶^{Σ^{l}D_{2}(i)_{∗}} π_{2m}(Σ^{l}D_{2}(X))

induced by the inclusion i: S^{m−l} ⟶ X of the bottom cell of X is surjective provided 2m < 2m + m_{1} − l, which is guaranteed by the assumption l ≤ m_{1} − 1. This proves claim 1.

For the proof of claim 2 we note that by Lemma 4.9 and Remark 4.10 we can identify the inclusion T ((m − l)H_{l}) ⟶ T ((m − l)H) with the inclusion ℝℙ_{m−l}^{m} ⟶ ℝℙ_{m−l} of stunted projective spaces. Our assumption l > 0 implies

H_{m}(ℝℙ_{m−l}; ℤ) = \{ \begin{matrix}ℤ/2 & m odd \\ 0 & m even \end{matrix} .

The fact that s′: S^{m} ⟶ ℝℙ_{m−l} induces a non-trivial map in integral homology then implies that m is odd and that for any lift s the composition p ∘ s has odd degree. Let g: S^{m} ⟶ ℝℙ_{m−l}^{m} be the attaching map for the top cell of ℝℙ_{m−l}^{m+1}. We note that for m odd the composition S^{m} ⟶^{g} ℝℙ_{m−l}^{m} ⟶^{p} S^{m} has mapping degree 2. The long exact sequence of stable homotopy groups associated to the cofibration

S^{m} ⟶^{g} ℝℙ_{m−l}^{m} ⟶^{i} ℝℙ_{m−l}^{m+1}

then shows that s′ ∈ π_{m}^{s}(ℝℙ_{m−l}) ≅ π_{m}^{s}(ℝℙ_{m−l}^{m+1}) can be factored through some s whose composition with p has mapping degree one. □

To prove Proposition 5.2 we will show the stronger statement that the Hopf invariant of the composition

(5.3)  g: S^{2m} ⟶^{c} M ⟶^{p_{1}∧p_{2}} (M_{1} ∧ M_{2})^{(2m)} ⟶^{h} K_{m}

is non-trivial. Here K_{m} is the Eilenberg MacLane space K_{m} = K(ℤ/2, m) (which up to homotopy equivalence is characterized by π_{m}(K_{m}) = ℤ/2, and the vanishing of all other homotopy groups), and h is the map corresponding to the generator of H^{m}((M_{1} ∧ M_{2})^{(2m)}; ℤ/2) ≅ ℤ/2. We recall that for any space X the map

[X, K_{m}] ⟶ H^{m}(X, ℤ/2) x ↦ x^{∗}(ι)

is an isomorphism. Here [M, K_{m}] is the group of homotopy classes of maps to K_{m}, and ι ∈ H^{m}(K_{m}; ℤ/2) = ℤ/2 is the generator. We use this isomorphism to identify cohomology classes with (homotopy classes of) maps to K_{m}.

For the proof that the Hopf invariant of the map (5.3) is non-trivial, it is useful to consider the following more general situation.

5.4 Definition. Let M be a framed manifold of dimension 2m, and let c: S^{2m} ⟶ M be its stable Thom map. Let

φ: H^{m}(M; ℤ/2) ⟶ ℤ/2

be the map which sends a map x: M ⟶ K_{m} to the Hopf invariant of the stable map

S^{2m} ⟶^{c} M ⟶^{x} K_{m}.

We note that H(x ∘ c) is an element of π_{2m}(D_{2}(K_{m})) = ℤ/2 (cf. [Mil, table 10.3]).

The above construction of φ is a special case of a much more general construction by E. Brown [Br, Cor. 1.11] (we deal here with the special case where the Thom spectrum MG is the sphere spectrum, and consider manifolds instead of Poincaré complexes). Specializing Lemma 1.14 and Corollary 1.11 of [Br] to our situation, we obtain the following properties for φ.

5.5 Proposition. Let M be a framed manifold of dimension 2m. Then φ has the following properties.

(1) If M is the boundary of a framed manifold W and x is in the image of the restriciton map H^{m}(W; ℤ/2) ⟶ H^{m}(M; ℤ/2) then φ(x) = 0.

(2) φ(x + y) = φ(x) + φ(y) + λ(x, y) for x, y ∈ H^{m}(M; ℤ/2). Here λ(x, y) = ⟨x ∪ y, [M]⟩ ∈ ℤ/2 is the intersection form ([M] ∈ H_{2m}(M; ℤ/2) is the mod 2 fundamental class of M).

Proof of Proposition 5.2. By the discussion before Definition 5.4 it suffices to show φ(x) ≠ 0, where x ∈ H^{m}(M; ℤ/2) is the cohomology class corresponding to the composition

M ⟶^{p_{1}∧p_{2}} (M_{1} ∧ M_{2})^{(2m)} ⟶^{h} K_{m}.

Let ā_{1} ∈ H^{m_{2}}(M_{1}; ℤ/2), ā_{2} ∈ H^{m_{1}}(M_{2}; ℤ/2), and b̄_{i} ∈ H^{m}(M_{i}; ℤ/2) be the mod 2 reduction of the integral cohomology classes a_{i}, b_{i} of section 3. We note that x is the product ā_{1}ā_{2} (as in section 3 we identify here H^{∗}(M_{i}) with its image under p_{i}^{∗} in H^{∗}(M)), which is equal to b̄_{1} + b̄_{2} by Corollary 3.4.

The class b̄_{i} is a restriction of a cohomology class in H^{m}(Dγ_{i}; ℤ/2). Hence λ(b̄_{i}, b̄_{i}) = 0 and φ(b̄_{i}) = 0 by Proposition 5.5(1). Since the intersection form λ is unimodular we get λ(b̄_{1}, b̄_{2}) = 1 and hence we conclude using part (2) of Proposition 5.5

φ(ā_{1}ā_{2}) = φ(b̄_{1} + b̄_{2}) = φ(b̄_{1}) + φ(b̄_{2}) + λ(b_{1}, b_{2}) = 1.

□

6. Desuspensions of (M_{1} ∧ M_{2})^{(2m)}

We recall that Proposition 2.7 claims that (M_{1} ∧ M_{2})^{(2m)} desuspends l times for some l with ϕ(l) ≥ ϕ(m_{1} − 1) except for (m_{1}, m_{2}) = (2, 2), (4, 5). In this section we prove Proposition 2.7 assuming (m_{1}, m_{2}) ≠ (8, 9), (8, 11); in the next section we will show that these cases don’t occur. Throughout the section we assume given a decomposition of S^{n+1} of the form (2.1) with 2 ≤ m_{1} < m_{2}, n = 2m, m = m_{1} +m_{2}.

We recall from section 3 that the assumption 2 ≤ m_{1} implies that the manifolds M_{1} and M_{2} are simply connected (Lemma 3.1) and that H^{q}(M_{i}; ℤ) = ℤ for q = 0, m_{i+1}, m, m + m_{i+1} and H^{q}(M_{i}; ℤ) = 0 for all other q (Lemma 3.3). It follows from obstruction theory that – up to homotopy equivalence – M_{i} allows a CW decomposition of the form

(6.1)  M_{i} = (S^{m_{i+1}} ∪_{f_{i}} e^{m}) ∪_{g_{i}} e^{m+m_{i+1}}

with attaching maps

f_{i} ∈ π_{m−1}(S^{m_{i+1}})  and  g_{i} ∈ π_{m+m_{i+1}−1}(S^{m_{i+1}} ∪_{f_{i}} e^{m}).

For the proof of Proposition 2.7 we need a number of lemmas.

6.2 Lemma. Let M_{i}^{(m)} = S^{m_{i+1}} ∪_{f_{i}} e^{m} be the m-skeleton of M_{i}. If M_{1}^{(m)} ∧ M_{2}^{(m)} desuspends l times with l ≤ m_{1}, then so does (M_{1} ∧ M_{2})^{(2m)}.

Proof. The CW-structures (6.1) on M_{1} resp. M_{2} induce a CW-structure on M_{1}∧M_{2} with three cells of dimension 2m; namely the m_{2}-cell in M_{1} times the top cell in M_{2}, the top cell in M_{1} times the m_{1}-cell in M_{2}, and the m-cell in M_{1} times the m-cell in M_{2}. We note that the first two of these cells are contained in the subcomplexes S^{m_{2}} ∧ M_{2} ⊂ M_{1} ∧ M_{2} and M_{1} ∧ S^{m_{1}} ⊂ M_{1} ∧ M_{2}, respectively, which obviously desuspend m_{1} times. Hence the attaching maps of these cells desuspend m_{1} times.

The third 2m-cell of M_{1} ∧ M_{2} is contained in the subcomplex M_{1}^{(m)} ∧ M_{2}^{(m)} ⊂ M_{1} ∧ M_{2}. Hence its attaching desuspends l times due to our assumption that M_{1}^{(m)} ∧ M_{2}^{(m)} desuspends l times. □

6.3 Lemma. ΣM_{i}^{(m)} is homotopy equivalent to the Thom space T (γ̄_{i+1}), where γ̄_{i+1} is the restriction of the vector bundle γ_{i+1} to S^{m_{i}} ⊂ M_{i+1}.

We remark that the image of the well-studied J-homomorphism consists precisely of the attaching maps of Thom spaces of vector bundles over spheres. In particular, Lemma 6.3 implies that Σf_{1} is in the image of the J-homomorphism.

Proof. The decomposition (2.1) implies that the quotient space S^{n+1}/ D(γ_{i}) is homeomorphic to the Thom space T (γ_{i+1}). The associated Puppe sequence

D(γ_{i}) ⟶ S^{n+1} ⟶ T (γ_{i+1}) ⟶ ΣD(γ_{i}) ⟶

shows that ΣD(γ_{i}) and hence ΣM_{i} is homotopy equivalent to T (γ_{i+1}) in dimensions ≤ n. This implies the lemma. □

6.4 Proof of Proposition 2.7 for m_{1} ≢ 0, 1, 2, 4 mod 8. We note that K̃O(S^{m_{1}}) is trivial for m_{1} ≢ 0, 1, 2, 4 mod 8 by Bott periodicity. This implies that the vector bundle γ̄_{2} (the restriction of γ_{2} to S^{m_{1}} ⊂ M_{2}) is trivial, since its fiber dimension is m_{2} + 1 > m_{1}. Hence its Thom space T (γ̄_{2}) is Σ^{m_{2}+1}S_{+}^{m_{1}}. It follows from Lemma 6.3 that ΣM_{1}^{(m)} desuspends (m_{2} + 1) times. This implies that M_{1}^{(m)} desuspends m_{2} times, since the condition m_{1} < m_{2} guarantees that the attaching map f_{1} ∈ π_{m−1}(S^{m_{2}}) lives in a stable group (i.e., Σ^{∞}: π_{m−1}(S^{m_{2}}) ⟶ π_{m−1}^{s}(S^{m_{2}}) is an isomorphism). □

In the cases m_{1} ≡ 0, 1, 2, 4 mod 8 we need the following lemma.

6.5 Lemma. Let γ be a vector bundle over S^{m_{1}}, m_{1} ≥ 9, of dimension m_{1} +k +1. Then the Thom space T (γ) desuspends k + 4 times.

Proof. Barratt and Mahowald prove that the map π_{q}(O(n)) ⟶ π_{q}(O) is surjective for n ≥ 13, q < 2(n − 1) [BM, Thm. 2]. Via the clutching construction which interprets elements of π_{q}(O(n)) as (isomorphism classes of) n-dimensional vector bundles over S^{q+1}, this implies that every vector bundle over S^{q+1} of dimension > q + 1 is obtained from an n-dimensional bundle by adding a trivial vector bundle of the appropriate dimension. This implies the lemma for m_{1} ≥ 16.

For m_{1} = 9, 10, 12 this follows from explicit calculations of Toda. We note that the Thom space T (γ) is a CW complex with one cell in dimension m_{1} + k + 1 and one cell in dimension 2m_{1} + k + 1, and hence its attaching map is an element of π_{2m_{1}+k}(S^{m_{1}+k+1}). By Toda’s results the suspension homomorphism

Σ^{k+4}: π_{2m_{1}−4}(S^{m_{1}−3}) ⟶ π_{2m_{1}+k}(S^{m_{1}+k+1})

is surjective for m_{1} = 9, 10, 12 (cf. Theorems 7.1 resp. 7.2 resp. 7.4 of [To]), and hence T (γ) desuspends k + 2 times. This proves the lemma (for m_{1} ≢ 0, 1, 2, 4 mod 8 the bundle γ is trivial thanks to Bott periodicity). □

6.6 Proof of Proposition 2.8 for m_{1} ≡ 0, 1, 2, 4 mod 8, (m_{1}, m_{2}) ≠ (8, 9), (8, 11).

First we assume m_{1} ≥ 9. In that case Lemma 6.5 implies that the Thom space T (γ̄_{2}) desuspends m_{2} − m_{1} + 4 times and hence M_{1}^{(m)} = Σ^{m_{2}−m_{1}+3}Y for some space Y by Lemma 6.3. In particular, (M_{1} ∧ M_{2})^{(2m)} desuspends at least 4 times by Lemma 6.2, and hence m + 1 is a multiple of 2^{ϕ(4)} = 8 by Corollary 2.9. It follows that m_{2} ≡ −m_{1} − 1 ≡ 7, 6, 5, 3 mod 8.

Now we write M_{1}^{(m)} ∧ M_{2}^{(m)} in the form

M_{1}^{(m)} ∧ M_{2}^{(m)} = Y ∧ Σ^{m_{2}−m_{1}+3}M_{2}^{(m)} = Y ∧ T (γ̄_{1} ⊕ \underline{ℝ}^{m_{2}−m_{1}+2}),

where \underline{ℝ}^{m_{2}−m_{1}+2} denotes the trivial vector bundle over S^{m_{2}} of dimension m_{2} − m_{1} + 2. We note that γ̄_{1} ⊕ \underline{ℝ}^{m_{2}−m_{1}+2} is a vector bundle over S^{m_{2}} of dimension m_{2} + 3. Since m_{2} ≢ 0, 1, 2, 4 mod 8, Bott periodicity implies that this vector bundle is trivial and hence its Thom space desuspends m_{2} + 3 times.

Now we study the cases m_{1} = 2, 4, 8. We note that by Freudenthal’s Suspension Theorem the Suspension homomorphism Σ^{∞}: π_{2m_{1}−1}(S^{m_{1}}) ⟶ π_{m_{1}}^{s} is surjective. In particular, the element f_{1} ∈ π_{m−1}(S^{m_{2}}), hence M_{1}^{(m)} and thus (M_{1} ∧ M_{2})^{(2m)} desuspend at least l = m_{2} − m_{1} times. We note that

(1) For m_{1} = 2 we have ϕ(m_{2} − m_{1}) ≥ ϕ(m_{1} − 1) = 1 provided m_{2} ≥ 3.

(2) For m_{1} = 4 we have ϕ(m_{2} − m_{1}) ≥ ϕ(m_{1} − 1) = 2 provided m_{2} ≥ 6.

(3) For m_{1} = 8 we have ϕ(m_{2} − m_{1}) ≥ ϕ(m_{1} − 1) = 3 provided m_{2} ≥ 12. This covers all cases with m_{1} < m_{2} except (m_{1}, m_{2}) = (4, 5) (which is excluded by assumption) and (m_{1}, m_{2}) = (8, 9), (8, 10), (8, 11). It’s easy to see that (8, 10) doesn’t occur: by the above argument (M_{1} ∧ M_{2})^{(2m)} desuspends l = m_{1} − m_{2} = 2 times and hence Corollary 2.9 leads to the contradiction that m + 1 = 19 is a multiple of 2^{ϕ(2)} = 4. □

7. Non-existence of multiplicities (m_{1}, m_{2}) = (8, 9), (8, 11)

In this section we show that there is no decomposition (2.1) of the sphere S^{n+1} with n = 2m, m = m_{1} + m_{2}, and (m_{1}, m_{2}) = (8, 9) or (m_{1}, m_{2}) = (8, 11). We keep the notation from the previous section; in particular, f_{i} and g_{i} are the attaching maps of the CW-decomposition (6.1) of M_{i}.

To rule out the case (8, 9), we need the following two lemmas.

7.1 Lemma. The composition

S^{m+m_{1}−1} ⟶^{Σ^{m_{1}} f_{1}} S^{m} ⟶^{Σf_{2}} S^{m_{1}+1}

is homotopic to the constant map.

Proof. Consider the Puppe sequence associated to the map f_{2}:

S^{m−1} ⟶^{f_{2}} S^{m_{1}} ⟶ S^{m_{1}} ∪_{f_{2}} e^{m} ⟶^{p} ΣS^{m−1} ⟶^{Σf_{2}} ΣS^{m_{1}} ⟶ .

Any two consecutive maps in this sequence have trivial composition; in particular, the composition Σf_{2} ∘ p ∘ g_{2} is trivial.

We note that the map p ∘ g_{2} ∈ π_{m+m_{1}−1}(S^{m}) describes how the top cell of M_{2} is attached to the m-cell. Since M_{2} is Spanier-Whitehead dual to M_{1}, this map is stably equivalent to f_{1}. This implies p ∘ g_{2} = Σ^{m_{1}}f_{1}, since the homotopy group π_{m+m_{1}−1}(S^{m}) is stable (i.e., the suspension homomorphism from this group to π_{m_{1}−1}^{s} is an isomorphism). This proves the lemma. □

7.2 Lemma.

(1) The index two subgroup of π_{7}^{s} ≅ ℤ/24 is in the image of Σ^{∞}: π_{14}(S^{7}) ⟶ π_{7}^{s}.

(2) Let m_{1} = 8, m_{2} = 9. If f_{1} ∈ π_{16}(S^{9}) ≅ π_{7}^{s} is an odd multiple of the generator, then ΣM_{2}^{(m)} desuspends at least twice.

Proof. The first statement follows from the EHP-sequence

⟶ π_{14}(S^{7}) ⟶^{Σ^{∞}} π_{14}^{s}(S^{7}) ⟶^{H} π_{14}(D_{2}(S^{7})) ⟶,

since π_{14}(D_{2}(S^{7})) = π_{14}(Σ^{7}ℝℙ_{7}) = H_{14}(Σ^{7}ℝℙ_{7}; ℤ) = ℤ/2.

To prove the second part, we need to show that Σf_{2} ∈ π_{17}(S^{9}) desuspends twice. Consider the following homotopy commutative diagram.

S^{22}  ⟶^{Σ^{6}f_{1}} S^{15}  ⟶^{f′_{2}} F_{2}C_{2}(S^{7}) ⟶^{p} D_{2,2}(S^{7}) ≈ Σ^{7}ℝℙ_{7}^{8}
i↓  ^{i}↓  ↓^{α_{2}}
Ω^{2}Σ^{2}S^{22} ⟶^{Ω^{2}Σ^{8}f_{1}} Ω^{2}Σ^{2}S^{15} ⟶^{Ω^{2}Σf_{2}} Ω^{2}Σ^{2}S^{7}

Here f′_{2} is a factorization up to homotopy of Ω^{2}Σf_{2} ∘ i through the map α_{2} which is a 20-equivalence by Theorem 4.4. Suppose that Σf_{2} does not desuspend twice or, equivalently, that f′_{2} does not factor through the inclusion i: S^{7} ⟶ F_{2}C_{2}(S^{7}). This implies by Corollary 4.6 that the composition p ∘ f′_{2} is non-trivial. We note that ℝℙ_{7}^{8} is the Moore space S^{7} ∪_{2} e^{8} obtained by attaching to S^{7} an 8-cell whose attaching map S^{7} ⟶ S^{7} has degree 2. This leads to the exact sequence

⟶ π_{q}^{s}(S^{7}) ⟶^{2} π_{q}^{s}(S^{7}) ⟶^{j_{∗}} π_{q}^{s}(ℝℙ_{7}^{8}) ⟶^{∂} π_{q−1}^{s}(S^{7}) ⟶^{2} π_{q−1}^{s}(S^{7}) ⟶, .

where j: S^{7} ⟶ ℝℙ_{7}^{8} is the inclusion map. Using π_{7}^{s}(S^{7}) ≅ ℤ and π_{8}^{s}(S^{7}) ≅ ℤ/2 generated by η (the suspension of the Hopf map), the sequence shows that π_{8}^{s}(ℝℙ_{7}^{8}) is isomorphic to ℤ/2 generated by j_{∗}η.

Assuming that Σ^{∞}f_{1} = kσ ∈ π_{7}^{s}, where k is odd and σ is the generator, then the composition

p ∘ f′_{2} ∘ Σ^{6}f_{1} ∈ π_{22}(Σ^{7}ℝℙ_{7}^{8}) = π_{15}^{s}(ℝℙ_{7}^{8})

is equal to kj_{∗}(η σ). This is a non-trivial element since ησ ∈ π_{8}^{s} is a non-trivial element not divisible by 2 (cf. [To, Thm. 7.1 and equation 7.4]).

On the other hand, the projection map p: F_{2}C_{2}(S^{7}) ⟶ D_{2,2}(S^{7}) factors stably in the form F_{2}C_{2}(S^{7}) ⟶^{α_{2}} Ω^{2}Σ^{2}S^{7} ⟶ D_{2,2}(S^{7}) by a result due to Snaith [Sn].

This factorization, and the fact that the composition Σf_{2} ∘ Σ^{8}f_{1} is trivial (by Lemma 7.1) show that p ∘ f′_{2} ∘ Σ^{6}f_{1} is stably trivial, which is the desired contradiction. □

7.3 Non-existence of (m_{1}, m_{2}) = (8, 9). If f_{1} ∈ π_{16}(S^{9}) ≅ ℤ/24 is an even multiple of the generator, then part (1) of Lemma 7.2 implies that f_{1} desuspends twice; hence so does M_{1}^{(m)}. If f_{1} is an odd multiple of the generator, then f_{1} desuspends once by Freudenthal’s Suspension Theorem; hence M_{1}^{(m)} = ΣX, and so M_{1}^{(m)} ∧ M_{2}^{(m)} = X ∧ ΣM_{2}^{(m)} desuspends twice by part (2) of Lemma 7.2.

In either case, M_{1}^{(m)} ∧ M_{2}^{(m)} desuspends at least twice, and hence so does (M_{1} ∧ M_{2})^{(2m)} by Lemma 6.2. Then Corollary 2.9 leads to the contradictory statement that m + 1 = 18 is a multiple of 2^{ϕ(2)} = 4. □

Now we turn to the case (m_{1}, m_{2}) = (8, 11). The first part of Lemma 7.2 shows that if f_{1} ∈ π_{18}(S^{11}) ≅ π_{7}^{s} ≅ ℤ/24 is an even multiple of the generator, then it is in the image of

Σ^{4}: π_{14}(S^{7}) ⟶ π_{18}(S^{11}).

Hence M_{1}^{(m)} desuspends 4 times and so does (M_{1} ∧ M_{2})^{(2m)} by Lemma 6.2. This leads by Corollary 2.9 to the contradictory statement that m + 1 = 20 is a multiple of 2^{ϕ(4)} = 8.

Hence from now on we will assume that f_{1} is an odd multiple of the generator. We recall that the mod 2 Hopf invariant is the homomorphism H: π_{q}^{s} ⟶ ℤ/2 which sends a stable map f: S^{k+q} ⟶ S^{k} to the non-trivial element if and only if the Steenrod operation Sq^{q+1} acts non-trivially on the cohomology of the mapping cone of f. E.g., Sq^{8} acts non-trivially on the cohomology of the Cayley plane, which shows that its attaching map has non-trivial mod 2 Hopf invariant. In particular, H: π_{7}^{s} ⟶ ℤ/2 is non-trivial. This implies H(f_{1}) ≠ 0; in other words,

(7.4)  Sq^{8}: H^{11}(X) ⟶ H^{19}(X)

is non-trivial, where X is the mapping cone of f_{1}; i.e., X = S^{11} ∪_{f_{1}} e^{19} = M_{1}^{(19)}. We recall from (6.1) that M_{1} = X ∪_{g_{1}} e^{30}.

7.5 Lemma. The attaching map g_{1} is stably trivial, i.e., it maps to zero under the homomorphism

Σ^{∞}: π_{29}(X) ⟶ π_{29}^{s}(X).

Proof. We note that the reduced cohomology of M_{1} is trivial except in dimensions 11, 19, and 30. This shows that the E_{2}-term

E_{2}^{p,q}(M_{1}) = H̃^{p}(M_{1}; KO^{q})

of the Atiyah-Hirzebruch spectral sequence converging to K̃O^{p+q}(M_{1}) is trivial for p + q = 0, since KO^{q} is trivial for −q ≢ 0, 1, 2, 4 mod 8. Hence any vector bundle over M_{1} of large enough dimension is trivial. In particular, the normal bundle of M_{1} in a sufficiently high dimensional euclidean space is trivial, and it is well known that this implies that the attaching map of the top cell is stably trivial.

For the convenience of the reader we recall the argument: a closed n-manifold M is Spanier-Whitehead dual to the Thom space T (γ), where γ is the normal bundle of some embedding of M into euclidean space. The projection map onto the top cell of M is dual to the inclusion of the bottom cell of T (γ). If γ has dimension k, the bottom cell of T (γ) has dimension k, and the inclusion of the bottom cell i: S^{k} ⟶ T (γ) is given geometrically by thinking of S^{k} as the Thom space of γ restricted to the base point. A trivialization of γ can be used to construct a map s: T (γ) ⟶ S^{k} which is a left inverse to i (i.e., the composition s ∘ i is the identity). Dually, there is a stable map S^{n} ⟶ M which is a right inverse to the projection map onto the top cell. This implies that the attaching map of the top cell is stably trivial. □

We recall from Corollary 4.6 (applied for k = ∞; cf. Remark 4.10) that the cofibration sequence

(7.6)  X ⟶^{i} F_{2}C(X) ⟶^{p} D_{2}(X)

is a homotopy fibration in the metastable range q < 3l−2 = 31, since the bottom cell of X is in dimension l = 11. We also recall that i_{∗}: π_{q}(X) ⟶ π_{q}(F_{2}C(X)) ≅ π_{q}^{s}(X) is the stabilization map Σ^{∞} for q < 3l − 1 = 32 and that the homotopy exact sequence of the fibration (7.6) is the EHP sequence. Hence Lemma 7.5 implies:

7.7 Corollary. There is an element g̃_{1} ∈ π_{30}(D_{2}(X)) which maps to g_{1} under the boundary map ∂: π_{30}(D_{2}(X)) ⟶ π_{29}(X) of the fibration (7.6).

7.8 Lemma. The map g̃_{1} induces a non-trivial homomorphism on reduced cohomology with ℤ/2-coefficients.

Proof. Consider the commutative diagram

X  X
i↓  ^{i}↓
X ∪_{g_{1}} e^{30} ⟶^{f} F_{2}C(X)
p↓  ^{p}↓
S^{30}  ⟶^{g̃_{1}} D_{2}(X)

Here the map f is constructed on e^{30} as a lift of D^{30} ⟶^{pr} D^{30}/S^{29} = S^{30} ⟶^{g_{1}} D_{2}(X); the restriction of this lift to S^{29} ⊂ D^{30} is g_{1} = ∂g̃_{1} (by definition of the boundary map), and hence the lift and the map i: X ⟶ F_{2}C(X) fit together to define the map f.

Let x_{i} ∈ H^{i}(M_{1}) ≅ H^{i}(X), i = 11, 19, be non-trivial elements (all cohomology groups here have ℤ/2 as coefficients, and since we work in the category of pointed spaces, we write H^{∗}(X) for the reduced cohomology of a pointed space X). The exact sequence

H^{i}(D_{2}(X)) ⟶^{p^{∗}} H^{∗}(F_{2}C(X)) ⟶^{i^{∗}} H^{∗}(X) ⟶^{∂} H^{i_{1}}(D_{2}(X)) ⟶ .

of the cofibration (7.6) implies that there are elements y_{i} ∈ H^{i}(F_{2}C(X)) with i^{∗}(y_{i}) = x_{i}. Again the exact sequence shows that there is an element y_{30} ∈ H^{30}(D_{2}(X)) with p^{∗}(y_{30}) = y_{11}y_{19}. Poincaré duality for M_{1} = X ∪_{g_{1}} e^{30} shows that the cup product x_{11}x_{19} ∈ H^{30}(M_{1}) is non-trivial. This implies by the commutative diagram above that g̃_{1}^{∗}(y_{30}) is non-trivial.

7.9 Lemma. For every map g: S^{30} ⟶ D_{2}(X) the induced homomorphism on reduced cohomology with ℤ/2-coefficients is trivial.

Clearly, the statements of Lemma 7.8 and 7.9 contradict each other. This shows that there can be no decomposition (2.1) of the sphere with (m_{1}, m_{2}) = (8, 11).

7.10 Remark. It might be instructive to discuss why the above arguments don’t work to exclude the possibilities (m_{1}, m_{2}) = (2, 2), (4, 5) (we recall from the introduction that there are isoparametric hypersurfaces with these multiplicities). In these cases it is not true that the normal bundle of M_{1} is stably trivial, and so Lemma 7.5 does not hold; if it did, we could prove non-existence of (2, 2) and (4, 5) the same way as we do for (8, 11).

Proof. We need to describe the ℤ/2-cohomology of the space D_{2}(X) (cf. [Mil, p. 40], [Jo, p. 256]). For the moment, let X be a CW complex, and x, y ∈ H^{∗}(X) be elements of degree p and q, respectively. Then there are elements in the cohomology of D_{2}(X) denoted

e^{t} ⊗ x ⊗ x ∈ H^{t+2p}(D_{2}(X)), t ≥ 0  and  [x, y] ∈ H^{p+q}(D_{2}(X)).

Moreover, if {x_{1}, x_{2}, … } is a basis of homogeneous elements of H^{∗}(X), then a basis for H^{∗}(D_{2}(X)) is provided by the following elements:

e^{t} ⊗ x_{i} ⊗ x_{i}, t ≥ 0 [x_{i}, x_{j}], i ≠ j.

Furthermore, the action of the Steenrod algebra on H^{∗}(D_{2}(X)) is determined by the following formulae for x ∈ H^{n}(X), y ∈ H^{m}(X), k ≥ 1:

\begin{aligned}Sq^{t}(e^{k} ⊗ x ⊗ x) & = ∑_{0≤i<t/2} \binom{k + n − i}{t − 2i}e^{k+t−2i} ⊗ Sq^{i}x ⊗ Sq^{i}x \\ Sq^{t}(e^{0} ⊗ x ⊗ x) & = ∑_{0≤i<t/2} \binom{n − i}{t − 2i}e^{t−2i} ⊗ Sq^{i}x ⊗ Sq^{i}x + ∑_{0≤i<t/2}[Sq^{t−i}x, Sq^{i}x] \\ Sq^{t}[x, y] & = ∑_{0≤i≤t}[Sq^{t−i}x, Sq^{i}y]. \end{aligned}

Now we specialize to X = M_{1}^{(m)} = S^{11} ∪_{f_{1}} e^{19}. A basis of H^{∗}(X) is {x_{11}, x_{19}}, where x_{11}, x_{19} are elements of degree 11 and 19, respectively. Hence H^{30}(D_{2}(X)) is generated by the elements [x_{11}, x_{19}] and e^{8} ⊗ x_{11} ⊗ x_{11}. Since Sq^{8}(x_{11}) = x_{19} (cf. (7.4)), the above formulae imply

\begin{aligned}Sq^{8}(e^{0} ⊗ x_{11} ⊗ x_{11}) & = e^{8} ⊗ x_{11} ⊗ x_{11} + [x_{19}, x_{11}] \\ Sq^{4}(e^{4} ⊗ x_{11} ⊗ x_{11}) & = e^{8} ⊗ x_{11} ⊗ x_{11} \end{aligned}

This shows that every element of H^{30}(D_{2}(X)) can be written as a linear combination of the elements Sq^{8}(e^{0} ⊗ x_{11} ⊗ x_{11}) and Sq^{4}(e^{4} ⊗ x_{11} ⊗ x_{11}). We note that the homomorphism induced by any map g: S^{30} ⟶ D_{2}(X) sends these elements to zero; e.g.,

g^{∗}(Sq^{8}(e^{0} ⊗ x_{11} ⊗ x_{11}) = Sq^{8}(g^{∗}(e^{0} ⊗ x_{11} ⊗ x_{11})) = 0

since g^{∗}(e^{0} ⊗ x_{11} ⊗ x_{11}) ∈ H^{22}(S^{30}) = 0. This proves Lemma 7.9. □

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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA and Max–Planck–Institut füer Mathematik, Gottfried–Claren-Str. 26, 53225 Bonn, Germany