FUKUMOTO-FURUTA INVARIANTS OF PLUMBED HOMOLOGY 3–SPHERES

NIKOLAI SAVELIEV

Abstract. Recent major progress in study of the homology cobordism group of homology 3-spheres is related to the work of Fukumoto, Furuta, and Ue on their w–invariant. We identify the w–invariant with the μ̄–invariant of Neumann and Siebenmann for all plumbed homology spheres. This leads to proving that all Seifert fibered and some classes of plumbed homology spheres with non-trivial Rohlin invariant have infinite order in the homology cobordism group.

The homology cobordism group Θ^{3} of oriented integral homology 3-sphe-res plays an outstanding role in the study of manifolds. One of its most spectacular applications is to the question of whether a closed topological manifold M of dimension five or higher is a polyhedron. It is shown in [9] and [10] that every such M is a polyhedron if and only if there exists a homology sphere with non-trivial Rohlin invariant which has second order in Θ^{3}. It is not known if such a homology sphere exists; in fact, it is not known whether Θ^{3} has any torsion at all.

The group Θ^{3} has been studied successfully in the last two decades by the methods of gauge theory. The main focus of such studies has been on Seifert fibered homology spheres Σ(a_{1}, …, a_{n}) and other graph homology spheres. First, Fintushel and Stern [2] have shown that certain manifolds Σ(a_{1}, …, a_{n}) have infinite order in Θ^{3}. Then Furuta [7] has found an infinite collection of Seifert fibered homology spheres of infinite order all of which are linearly independent in Θ^{3}, thus effectively establishing that Θ^{3} is an infinitely generated abelian group.

Gauge theoretical methods have also given a boost to the study of Θ^{3} via the μ̄–invariant of Neumann [11] and Siebenmann [17]. It has produced more examples of Seifert fibered homology spheres having infinite order in

Research was supported in part by NSF Grant DMS-9896376 and Max-Planck-Institut für Mathematik in Bonn.

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Θ^{3}, see [14] and [15]. Of particular importance in this study was the so called 10/8–condition on the signature of smooth spin manifolds established by Furuta [8]. Recently, Fukumoto and Furuta [5] extended this result to spin V–manifolds. Given an integral homology sphere Σ, they introduced a family of invariants w(Σ, X, c) of Σ and a spin V–manifold (X, c) it bounds, and obtained some strong restrictions on the behavior of w(Σ, X, c) whenever Σ has finite order in Θ^{3}.

In the special case of Seifert fibered homology sphere Σ = Σ(a_{1}, …, a_{n}), Fukumoto, Furuta and Ue in [6] showed that w(Σ, X, c) = −μ̄(Σ) for a specific choice of (X, c). Based on this result, they proved that μ̄ is a homology cobordism invariant for all Σ(a_{1}, …, a_{n}) and their multiples provided one of Seifert invariants a_{i} is even, or otherwise their number does not exceed six.

We develop a general theory of the w–invariant for plumbed homology spheres and plumbed V–manifolds. Plumbed V–manifolds comprise a rather large class of V–manifolds and include, in particular, all V–manifolds studied in [3] and [6] in connection with plumbed homology spheres. Every plumbed homology sphere bounds a spin plumbed V–manifold.

Theorem 1. Let Σ be a plumbed homology sphere then w(Σ, X, c) = −μ̄(Σ) for every spin plumbed V–manifold (X, c) such that ∂X = Σ.

Using this theorem, we study homology cobordism properties of Seifert fibered and plumbed homology spheres. We notice that the results of [6] and [15] can be combined to remove the above mentioned restriction on Seifert fibered homology spheres of [6], and prove the following general result.

Theorem 2. Any Seifert fibered homology sphere Σ(a_{1}, …, a_{n}) with nonvanishing μ̄–invariant has infinite order in the homology cobordism group.

The modulo 2 reduction of the μ̄–invariant coincides with the Rohlin invariant. In particular, if the Rohlin invariant of Σ(a_{1}, …, a_{n}) is nontrivial then Σ(a_{1}, …, a_{n}) cannot have order two in Θ^{3}. Thus if one is to look for an element in Θ^{3} solving positively the triangulation conjecture, all Seifert fibered homology spheres, as well as all homology spheres them homology cobordant, should be excluded from consideration.

HOMOLOGY 3–SPHERES 3

The methods of [5] and [6] are further applied to study homology cobordism properties of general plumbed homology spheres. It is not clear if these techniques are strong enough to prove that μ̄ is a homology cobordism invariant for all of them. However, we use them to construct an infinite family of plumbed homology spheres which are not Seifert fibered but for which the conclusion of Theorem 2 is true. This is a generalization of the results of [3].

Theorem 3. Let Σ be the splice of two Seifert fibered homology spheres Σ(a_{1}, …, a_{n}) and Σ(α_{1}, …, α_{m}) along fibers of degrees a_{n} and α_{m}, and let p = a_{1} ⋯ a_{n−1} · α_{1} ⋯ α_{m−1} − a_{n} · α_{m}. Suppose that Σ has finite order in the homology cobordism group. If p ≥ 0 then μ̄(Σ) = 0, if p < 0 then μ̄(Σ) is either zero or one.

Note that Σ is an algebraic link if and only if p ≤ 0, and it is Seifert fibered if p = 0. Thus the theorem does not provide us with a definite conclusion about μ̄(Σ) precisely in the case when Σ is a non-Seifert fibered algebraic links.

I wish to thank Dr. Furuta for useful discussions at Max-Planck-Institut für Mathematik in Bonn in 1998, and Dr. Fukumoto and Dr. Ue for sending me preprints [3], [4] and [6].

1. The w–invariants of Fukumoto and Furuta

In this section we review the w–invariants defined in [5] by using V– manifolds. We formulate Fukumoto-Furuta’s extension of the smooth 10/8– theorem to spin V–manifolds and show how it is related to certain homology cobordism properties of the w–invariants.

1.1. Closed V–manifolds. The concept of V–manifold [13] will not be used in this paper in its full generality. We will only deal with compact 4-dimensional V–manifolds with isolated singular points. If X is such a V-manifold, each its singular point has a neighborhood isomorphic to the cone over a lens space or other spherical space form. The usual concepts of topology and differential geometry extend to V–manifolds after defining everything in an equivariant setting. In particular, we will be interested in a spin structure on W and the (complex) index of the associated Dirac

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operator D(X). Fukumoto and Furuta proved in [5] the following result about this index.

Theorem 4. Let X be an oriented closed spin V–manifold with b_{1}(X) = 0, and let c be a spin structure on X. Then ind D(X) is even, and either ind D(X) = 0 or b_{−}(X) < ind D(X) < b_{+}(X).

Note that if X is actually smooth, the statement of this theorem is equivalent to the famous 10/8–theorem of Furuta [8]. In this case, the index of Dirac operator is computed from the index theorem as

ind D(X) = −p_{1}(X)/24 = − sign(X)/8,  (1)

which together with inequality b_{−}(X) < ind D(X) < b_{+}(X) of the theorem implies that b_{2}(X)/| sign(X)| > 10/8.

1.2. Definition of w–invariants. Let Σ be an oriented integral homology sphere, and let X be a compact oriented spin V–manifold with only isolated singularities such that b_{1}(X) = 0 and ∂X = Σ. Such a V–manifold always exists, moreover, it can be chosen to be smooth and simply connected. Let c be a spin structure on X. For every triple (Σ, X, c), the w–invariant is defined in [5] as

w(Σ, X, c) = ind D(X ∪ Y) + sign(Y)/8,

where Y is any smooth spin 4–manifold with ∂Y = −Σ. Here, D(X ∪ Y) is Dirac operator associated with the spin structure on X ∪ Y induced by c and the spin structure on Y. Of course, if X is smooth we simply get w(Σ, X, c) = − sign(X)/8.

The definition of w(Σ, X, c) does not depend on the choice of Y. The reduction of w(Σ, X, c) modulo 2 equals the Rohlin invariant of Σ for any choice of X and c. Moreover, w(Σ, X, c) is additive with respect to connected sums of the triples (Σ, X, c). All these statements are verified in [5]. Together with Theorem 4 they imply the following important property of the w– invariants.

Theorem 5. Let Σ be an oriented integral homology sphere which has finite order in the homology cobordism group Θ^{3}. Then, for any choice of X and c, either w(Σ, X, c) = 0 or

−b_{−}(X) < w(Σ, X, c) < b_{+}(X).

HOMOLOGY 3–SPHERES 5

For example, if Σ happens to bound a spin V–manifold X such that b_{+}(X) ≤ 1, b_{−}(X) ≤ 1 and w(Σ, X, c) ≠ 0, then Σ cannot be of finite order in the homology cobordism group. The strength of Theorem 5 lies in the fact that many homology spheres Σ do bound spin V–manifolds X with small values of b_{±}(X).

2. Calculus of plumbing graphs

This section is devoted to 3- and 4-dimensional manifolds obtained by plumbing D^{2}–bundles over 2-spheres, and to the μ̄–invariant of Neumann and Siebenmann.

2.1. Plumbed manifolds. Plumbing graph Γ is a graph with no cycles, each of whose vertices v_{i} carries an integer weight e_{i}, i = 1, …, s. Associate to each vertex v_{i} the D^{2}–bundle Y (e_{i}) over S^{2} with the Euler number e_{i}. If the vertex v_{i} has d_{i} edges connected to it in the graph Γ, choose d_{i} disjoint discs in the base of Y (e_{i}) and call the disc bundle over the j–th disc B_{ij}, so that B_{ij} = D^{2}×D^{2}. When two vertices, v_{i} and v_{k}, are connected by an edge, identify B_{ij} with B_{kℓ} by exchanging the base and fiber coordinates. This pasting operation is called plumbing, and the resulting smooth 4-manifold P (Γ) is said to be obtained by plumbing according to Γ. Its boundary M(Γ) = ∂P (Γ) is usually referred to as a plumbed 3–manifold.

The 4–manifold P (Γ) is simply connected. The group H_{2}(P (Γ)) has a natural basis represented by the zero-sections of the plumbed bundles. We note that all these sections are embedded 2-spheres, and that they can be oriented in such a way that the intersection form of P (Γ) will be given by the matrix A(Γ) = (a_{ij})_{i,j=1,... ,s} with the entries

a_{ij} = \begin{cases}e_{i}, & if i = j; \\ 1, & if i is connected to j by an edge; \\ 0, & otherwise. \end{cases}

Let M = ∂P (Γ) be a 3-dimensional manifold obtained by plumbing according to a graph Γ, then M is an integral homology sphere if and only if the matrix A(Γ) is unimodular, that is det A(Γ) = ±1.

Different plumbing graphs can describe the same 3-dimensional manifold. In fact, plumbing graphs Γ_{1} and Γ_{2} describe the same 3-manifold if and only if Γ_{1} can be obtained from Γ_{2} by a sequence of moves described in [1] and in [17].

2.2. Seifert manifolds and lens spaces. Let M be a 3-manifold Seifert fibered over S^{2} with Seifert invariants (b; (a_{1}, b_{1}), …, (a_{n}, b_{n})), a_{i} ≥ 1. It can be thought of as the boundary of the 4-manifold obtained by plumbing according to the star-shaped graph shown in Figure 1.

PSfrag replacements t_{11}

t_{12}
t_{1m_{1}}
t_{21}
t_{22}
t_{2m_{2}}
t_{n1}
t_{n2}
t_{nm_{n}}
b

Figure 1

The integer weights t_{ij} in the graph are found from continued fractions a_{i}/b_{i} = [t_{i1}, …, t_{im_{i}}] where i = 1, …, n and we use the notation

[t_{1}, …, t_{k}] = t_{1} − \frac{1}{t_{2} − \frac{1}{⋯ − \frac{1}{t_{k}}}}

Different decompositions of a_{i}/b_{i} into continued fractions lead to plumbing graphs describing the same 3–manifold M but different 4–manifolds P (Γ).

Seifert manifold M is an integral homology sphere if and only if the determinant of matrix A(Γ) associated with the plumbing graph in Figure 1 is ±1, which translates into the condition

a_{1} ⋯ a_{n} · (−b + ∑_{i=1}^{n} b_{i}/a_{i}) = ±1.  (2)

We fix an orientation on M by choosing +1 in (2). By considering (2) modulo a_{i} we see that it determines b_{i} modulo a_{1} which is enough to determine the manifold M completely. We use the notation Σ(a_{1}, …, a_{n}) for this manifold and call it a Seifert fibered homology sphere.

In general, the order of the first homology of M(b; (a_{1}, b_{1}), …, (a_{n}, b_{n})) equals the absolute value of

a_{1} ⋯ a_{n} · (−b + ∑_{i=1}^{n} b_{i}/a_{i}) ,

unless the latter is equal to zero, in which case H_{1}(M, ℤ) is infinite.

Lens spaces are a special case of Seifert fibered manifolds. Our orientation convention will be that a lens space L(p, q) is obtained by (−p/q)– Dehn surgery on an unknot in S^{3}. Expanding −p/q = [t_{1}, …, t_{n}] in a continuous fraction we get L(p, q) as the boundary of the 4–manifold obtained by plumbing on the chain Γ shown in Figure 2. We will use notation A(Γ) = Δ(t_{1}, …, t_{n}).

PSfrag replacements

t_{1}
t_{2}
t_{n}

Figure 2

One can notice that this plumbing graph also represents lens space L(p, q^{∗}) with −p/q^{∗} = [t_{n}, …, t_{1}] where qq^{∗} = 1 mod p. This reflects the fact that L(p, q) and L(p, q^{∗}) are homeomorphic. Our convention will be that we will include S^{3} but not S^{1} × S^{2} into lens spaces : these can both be obtained by chain plumbing, see for instance Figure 3. Thus our convention will be that all lens spaces are rational homology spheres.

PSfrag replacements 0 1 −7 2 S^{3}

S^{1} × S^{2}

Figure 3

2.3. Intersection forms of plumbed 4–manifolds. There is a simple algorithm to diagonalize matrices of the form A(Γ), see [1]. This algorithm is a graph realization of the usual Gauss diagonalization process for symmetric matrices. Let us allow trees weighted by arbitrary rational numbers. Given such a tree, pick a vertex and direct all its edges toward this vertex. Now,

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the tree can be simplified by performing operations of the two types shown in Figures 4 and 5 (we re-index vertices if necessary).

PSfrag replacements e_{j} e_{1} e_{2} e_{k} e′_{j} e_{k+1} 0 1 −1

e′_{j} = e_{j} − \frac{1}{e_{1}} − … − \frac{1}{e_{k}}

Figure 4 PSfrag replacements

e_{j} e_{1} e_{2} e_{k} e′_{j} e_{k+1} 0 1 −1

Figure 5

We end up with a finite collection of isolated points weighted by rational numbers d_{1}, …, d_{s}. Then D = diag (d_{1}, …, d_{s}) is a diagonalization of A(Γ), that is, A(Γ) = U^{⊤}DU with det U = ±1. In particular, det A(Γ) = det D and sign A(Γ) = sign D.

Example. For the tree Γ in Figure 6 we have det A(Γ) = det D = 2 · 3 · (1/6) · (−2) · (−1/2) = 1 and sign A(Γ) = sign D = 3 − 2 = 1.

PSfrag replacements 1 2 3 5 −2 1/6 −1/2

HOMOLOGY 3–SPHERES 9

Example. The diagonalization of the matrix A(Γ) = Δ(t_{1}, …, t_{n}) corresponding to the tree in Figure 2 directed from right to left has entries

[t_{n}], [t_{n−1}, t_{n}], …, [t_{2}, …, t_{n−1}, t_{n}], [t_{1}, …, t_{n−1}, t_{n}].

The determinant of Δ(t_{1}, …, t_{n}) equals (up to a sign) the numerator of the reduced fraction [t_{1}, …, t_{n−1}, t_{n}]. We extend the concept of continued fraction by allowing division by zero in the sense of Figure 5.

2.4. Graph links and splicing. By a link (Σ, K) = (Σ, k_{1} ∪ … ∪ k_{m}) we will mean a pair consisting of an oriented integral homology 3-sphere Σ and a collection K of disjoint oriented knots S_{1}, …, S_{m} in Σ. Empty links are also allowed; they are just homology spheres. A link (Σ(a_{1}, …, a_{n}), S_{1}∪…∪S_{m}) where the link components S_{1}, …, S_{m} are fibers in Σ(a_{1}, …, a_{n}) is called a Seifert link. Note that a general Seifert link may include singular fibers as well as some non-singular fibers.

Let (Σ, K) and (Σ′, K′) be links and choose components S ⊂ K and S′ ⊂ K′. Let N(S) and N(S′) be tubular neighborhoods and m, ℓ ⊂ ∂N(S) and m′, ℓ′ ⊂ ∂N(S′) be standard meridians and longitudes. The manifold

Σ′′ = (Σ \ int N(S)) ∪ (Σ′ \ int N(S′))

obtained by pasting along boundaries by matching m to ℓ′ and m′ to ℓ, is an integral homology sphere. The link (Σ′′, (K \ S) ∪ (K′ \ S′)) is called the splice of (Σ, K) and (Σ′, K′) along S and S′.

Any link which can be obtained from a finite number of Seifert links by splicing is called a graph link. Empty graph links are precisely plumbed, or graph, homology spheres, see [1]. Graph links can be conveniently represented by plumbing diagrams with arrowhead vertices attached to the boundary vertices corresponding to the link components.

Example. Figure 7 shows Seifert link (S^{3}, K) where K is the right handed trefoil, corresponding to a regular fiber in a Seifert fibration of S^{3}.

PSfrag replacements −2

−3
−1

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Example. Figure 8 shows Seifert link (Σ(2, 3, 7), K) with K representing the singular fiber of degree 7.

PSfrag replacements −2

−3
−1
−7

Figure 8

Splicing operation on graph links can be described in the language of plumbing graphs as follows. Suppose we are given two graph links represented by their plumbing diagrams Γ and Γ′ as shown in Figure 9 with arrows attached to vertices weighted by e_{n} and e′_{m} respectively.

PSfrag replacements Γ Γ′ e_{n} e′_{m}

Figure 9

The plumbing diagram for the spliced link is shown in Figure 10. Here, a = det A(Γ_{0})/ det A(Γ) where Γ_{0} is the portion of the plumbing graph Γ obtained by removing the n–th vertex weighted by e_{n} and all adjacent edges, and similarly for a′.

PSfrag replacements

a
a′
e_{n}
e′_{m}

Figure 10

Example. The diagrams in Figure 11 show plumbed homology spheres obtained by splicing two copies of S^{3} along left- and right-handed trefoils (on the left), and by splicing Σ(2, 3, 7) and Σ(2, 3, 11) along singular fibers of degrees 7 and 11 (on the right). Both diagrams were simplified by using the usual rules.

HOMOLOGY 3–SPHERES 11

PSfrag replacements 2 3 −2 −3 −8 0 5 −6 6 1 −1

Figure 11

The above description of splicing in terms of plumbing graphs allows us to think of it as an operation on the corresponding plumbed 4-manifolds. This splicing operation in four dimensions can be described as follows. Let W and W′ be plumbed 4-manifolds with boundaries ∂W = Σ and ∂W′ = Σ′, and let Σ′′ be the splice of Σ and Σ′ along knots S and S′. Represent Σ as the union Σ = X ∪ (T × I) ∪ N(S) where torus T is the boundary of a tubular neighborhood N(S) of S, X is the knot exterior, and I = [0, 1]. Let Σ′ = X′ ∪ (T × I) ∪ N(S′) in a similar fashion. Then we can glue W to W′ along T × I to get a manifold with boundary Σ′′ ∪ S^{3}, the S^{3} component coming from gluing N(S) with N(S′) by matching meridian to longitude and vice versa. Filling S^{3} with 4-dimensional ball, we get the same plumbed 4-manifold as from the operation on plumbing graphs.

2.5. The μ̄–invariant of plumbed homology spheres. For any plumbed homology sphere Σ = ∂P (Γ), there exists a unique homology class w ∈ H_{2}(P (Γ), ℤ) satisfying the following two conditions. First, w is characteristic, that is (dot represents intersection number)

w · x = x · x mod 2 for all x ∈ H_{2}(P (Γ), ℤ),  (3)

and second, all the coordinates of w are either 0 or 1 in the natural basis of H_{2}(P (Γ), ℤ). We call w the Wu class for P (Γ). It is proved in [11] that the integer

μ̄(Σ) = \frac{1}{8} (sign P (Γ) − w · w)  (4)

only depends on Σ and not on Γ. We call it the μ̄-invariant. One can easily see that condition (3) implies that for no two adjacent vertices in Γ the corresponding coordinates of w are both equal to 1. Therefore, the class w can be chosen to be spherical, and μ̄(Σ) = μ(Σ) mod 2 where μ(Σ) is the Rohlin invariant.

Another description of the μ̄–invariant is as follows. In the natural basis of H_{2}(P (Γ), ℤ), the defining property (3) translates into the linear system

∑_{i=1}^{s} a_{ij}w_{j} = a_{ii} mod 2, i = 1, …, s.  (5)

Due to the fact that det A(Γ) = 1 mod 2, this system has a unique solution w = (w_{1}, …, w_{s}) over ℤ/2. One can show that w = diag A(Γ)^{−1} mod 2, see for example [16]. Hence w_{i} can be computed as the mod 2 determinant of A(Γ_{i}) where Γ_{i} is obtained from Γ by removing vertex v_{i} and all its adjacent edges. The subset S of vertices v_{i} of Γ such that w_{i} = 1 mod 2 is called the Wu set of Γ. The μ̄–invariant is then given by the formula

μ̄(Σ) = \frac{1}{8} (sign P (Γ) − ∑_{v_{i}∈S} a_{ii}) .

The μ̄–invariant is easy to compute. It is additive with respect to splicing, so one only needs to compute it for Seifert fibered homology spheres. Explicit formulas for μ̄(Σ(a_{1}, …, a_{n})) can be found in [11].

All the above works as well for any plumbed ℤ/2–homology sphere M thus defining μ̄(M). It is no longer an integer although 8 · μ̄(M) still is. For plumbed 3–manifolds M which are not ℤ/2–homology spheres, one can define an invariant μ̄(M, c) which will depend on a spin structure c on M, see [11] and [16].

3. Plumbed V–cobordisms

In this section we take a different point of view on plumbed 4-manifolds and think of them as cobordisms between plumbed 3-manifolds and disjoint unions of lens spaces. We describe such cobordisms by decorated plumbing graphs.

3.1. Plumbed manifolds as V–cobordisms. Given a 4-manifold P (Γ) obtained by plumbing on a graph Γ let Γ′ be a chain in Γ of the form shown in Figure 2. Plumbing on Γ′ yields a submanifold P (Γ′) of P (Γ) with boundary a lens space L(p, q). The closure of P (Γ) \ P (Γ′) is a smooth compact 4-manifold X_{0} with oriented boundary −L(p, q) ∪ ∂P (Γ), which can be thought of as a cobordism between L(p, q) and ∂P (Γ). Starting with several chains Γ′ in Γ, one can establish a cobordism between ∂P (Γ) and a

HOMOLOGY 3–SPHERES 13

disjoint union L of several lens spaces. The chains Γ′ must be disjoint in the following sense : no two chains have a common vertex, and no edges of Γ have one endpoint on one chain and the other endpoint on the other.

Such cobordisms will be called plumbed V–cobordisms. A plumbed V– cobordism X_{0} is always a smooth manifold; the name V–cobordism refers to the fact that each lens space on the boundary of X_{0} can be capped off with a cone to make it into a V–manifold X. The latter will be referred to as a plumbed V–manifold.

Plumbed V–cobordisms can be conveniently represented by decorated plumbing graphs. These are plumbed graphs Γ with extra circles (not necessarily round) each of which encloses exactly one chain Γ′. The above condition on the chains to be disjoint translates into the condition that no edge of Γ should be intersected by more than one decorating circle.

PSfrag replacements 2 3 −2 −3 −8 0 5

Figure 12

Example. Consider the plumbing diagram in Figure 12 with circles as shown in Figure 12. It describes a plumbed V–cobordism X_{0} between L = L(2, −1) ∪ L(2, 1) ∪ L(41, 5) ∪ L(3, 1) ∪ L(3, −1) and Σ, where Σ is the splice of Σ(2, 3, 7) and Σ(2, 3, 11) along the singular fibers of degrees 7 and 11, compare with Figure 11. The only generators in the second homology of X_{0} come from the vertices outside all of the circles, that is the vertices weighted by 0. Therefore, H_{2}(W_{0}, ℤ) = ℤ ⊕ ℤ.

3.2. Rational intersection forms of plumbed V-cobordisms. By the rational intersection form of a plumbed V–cobordism X_{0} with ∂X_{0} = −L∪Σ we mean the cup-product pairing

B : H^{2}(X_{0}, ℚ) ⊗ H^{2}(X_{0}, ℚ) → ℚ .

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The form B is non-degenerate over the rationals whenever H_{1}(Σ, ℚ) = 0. The form B is also referred to as the rational intersection form of the plumbed V–manifold X, and by definition sign X = sign X_{0} = sign B.

Let X_{0} be represented by a decorated plumbing graph Γ so that P (Γ) = ⋃ P (Γ′) ∪_{L} X_{0} where Γ′ are disjoint chains inside the decorating circles. Since H_{1}(L, ℚ) = 0 we have a splitting

A(Γ) = ⨁ A(Γ′) ⊕ B.  (6)

In particular, sign A(Γ) = ∑ sign A(Γ′)+sign B and det A(Γ) = ∏ det A(Γ′)· det B. Let L = ⋃ L(p_{i}, q_{i}) and suppose that Σ is an integral homology sphere. Then det A(Γ) = ±1 and det B = ±1/p_{1} ⋯ p_{n}.

In practical terms, the intersection form B = (b_{ij}) can be calculated as follows. We call a vertex v and a decorating circle adjacent if the circle intersects an edge one of whose endpoints is v.

The generators in the second homology of X_{0} correspond to the vertices of Γ outside of all the decorating circles. Given such a vertex v_{i} weighted by an integer e_{i}, we have

b_{ii} = e_{i} − ∑ d_{k}, d_{k} = det A(Γ′_{k})/ det A(Γ_{k}).

Here, the summation goes over all the circles adjacent to v_{i}, Γ_{k} is the chain inside the k–th circle, and Γ′_{k} is the portion of Γ_{k} obtained by removing the vertex inside Γ_{k} adjacent to v_{i} and removing all its adjacent edges. We agree that the determinant of an empty graph is equal to 1.

For any two generating vertices v_{i} and v_{j} connected by an edge inside Γ away from the decorating circles, b_{ij} = 1. If these two vertices are adjacent to the same decorating circle, we have

b_{ij} = det A(Γ′_{ij})/ det A(Γ_{ij})

where Γ_{ij} is the chain inside the circle, and Γ′_{ij} is its portion obtained by removing all the vertices of Γ′_{ij} belonging to the shortest chain connecting v_{i} with v_{j}, and removing all their adjacent edges. Note that the sign of b_{ij}, i ≠ j, can be reversed by changing orientations of the generators v_{i} or v_{j}. Since our plumbing graphs have no cycles, any choice of signs for b_{ij}, i ≠ j, will do.

Example. The rational intersection form of the plumbed V–cobordism described by the decorated plumbing graph in Figure 12 is given by

B = (\begin{matrix}−11/246 & −1/41 \\ −1/41 & −7/246 \end{matrix}) .

Example. An important problem in calculations with plumbing graphs is to identify the extra lens space that arises from splicing, see Figure 13 where the lens space in question is circled. PSfrag replacements

u_{1} u_{k} 0

Figure 13

This problem can be solved as follows. In the canonical meridian-longitude bases, the splicing is given by the formula

(\begin{matrix}m \\ ℓ \end{matrix}) = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix}m′ \\ ℓ′ \end{matrix}) .

To find the parameters of the lens space, one should re-write this gluing operation in the base-fiber bases. Suppose that we splice Seifert links (Σ(a_{1}, …, a_{n}), S) and (Σ(α_{1}, …, α_{m}), S′) along Seifert fibers S and S′ of degrees a_{n} and α_{m}, respectively. Then

(\begin{matrix}m \\ ℓ \end{matrix}) = (\begin{matrix}a_{n} & b_{n} \\ −a_{1} ⋯ a_{n−1} & d_{n} \end{matrix}) (\begin{matrix}x \\ h \end{matrix}) , (\begin{matrix}m′ \\ ℓ′ \end{matrix}) = (\begin{matrix}α_{m} & β_{m} \\ −α_{1} ⋯ α_{m−1} & δ_{m} \end{matrix}) (\begin{matrix}x′ \\ h′ \end{matrix}) ,

where

d_{n} = a_{1} ⋯ a_{n−1} · ∑_{i=1}^{n−1} b_{i}/a_{i} and δ_{m} = α_{1} ⋯ α_{m−1} · ∑_{j=1}^{m−1} β_{j}/α_{j}.

Therefore,

(\begin{matrix}x \\ h \end{matrix}) = (\begin{matrix}q & −p^{∗} \\ −p & −q^{∗} \end{matrix}) (\begin{matrix}x′ \\ h′ \end{matrix}) ,

where p = a_{1} ⋯ a_{n−1} · α_{1} ⋯ α_{m−1} − a_{n} · α_{m},

q = −a_{1} ⋯ a_{n−1} · α_{1} ⋯ a_{m−1} · ∑_{i=1}^{n−1} b_{i}/a_{i} − b_{n} · α_{m},

q^{∗} = −a_{1} ⋯ a_{n−1} · α_{1} ⋯ a_{m−1} · ∑_{j=1}^{m−1} β_{j}/α_{j} − β_{m} · α_{n},

and p^{∗} is determined from equation pp^{∗} + qq^{∗} = 1 as

p^{∗} = −a_{1} ⋯ a_{n−1} · α_{1} ⋯ α_{m−1} ∑_{i=1}^{n−1} b_{i}/a_{i} ∑_{j=1}^{m−1} α_{j}/β_{j} + b_{n}β_{m}.

The resulting lens space is L(p, q) with the above parameters p and q. Note that it only depends on the two spliced Seifert links and not on the rest of the manifold.

3.3. Spin structures on plumbed V–cobordisms. There exists a simple algorithm which decides whether a given plumbed V–cobordism X_{0} admits a spin structure. If such a spin structure exists it has to be unique because H_{1}(X_{0}, ℤ) = 0.

Given a V–cobordism X_{0} represented by a decorated plumbing diagram Γ, consider the vertices outside all of the circles. They form a free basis in H_{2}(X_{0}, ℤ). Enumerate these vertices from 1 to n and let w_{i} = det A(Γ_{i}) mod 2 where Γ_{i} is obtained from Γ by removing the i–th vertex and all adjacent edges, i = 1, …, n.

Proposition 6. The V–cobordism X_{0} admits a spin structure if and only if w_{i} = 0 mod 2 for all i = 1, …, n.

Proof. We work in the above specified basis. The V–cobordism X_{0} is spin if and only if its integral intersection matrix on H_{2}(X_{0}, ℤ) is even. The latter is the inverse of the rational intersection matrix B. The i–th diagonal element of B^{−1} equals det B_{i}/ det B, where B_{i} is the minor obtained from B by removing the i–th row and i–th column.

To compute the determinants, let us introduce the following notation. The chains circled in the graph Γ will be denoted by Γ′_{j}. For every j, the

HOMOLOGY 3–SPHERES 17

lens space obtained by plumbing according to Γ′_{j} will be called L(p_{j}, q_{j}), so that p_{j} = ± det A(Γ′_{j}). The determinant of B can be calculated using the splitting (6) as

det B = \frac{det A(Γ)}{∏_{j} det A(Γ′_{j})} = ± \frac{1}{∏_{j} p_{j}} ,

where we take into account that det A(Γ) = ±1 since Σ is an integral homology sphere. Similarly,

det B_{i} = \frac{det A(Γ_{i})}{∏_{j} det A(Γ′_{j})}.

Therefore, w_{i} = det B_{i}/ det B = det A(Γ_{i}) mod 2, and the result follows.

Example. The plumbed V–cobordism X_{0} shown in Figure 12 is spin. This can be seen in two different ways. One is to find its rational intersection form B and its inverse,

B^{−1} = (\begin{matrix}−42 & 36 \\ 36 & −66 \end{matrix}) ,

and note that B^{−1} is even. The other is to apply Proposition 6 as follows. Enumerate the vertices generating H_{2}(X_{0}, ℤ) from left to right. Removing vertex number 1 together with its adjacent edges leaves us with the graph Γ_{1} shown in Figure 14.

PSfrag replacements 2 3 −2 −3 −8 0 5

Figure 14

The determinant of A(Γ_{1}) is the product of the determinants of the connected components of Γ_{1}. The one vertex component in the upper left corner has even determinant, therefore, w_{1} = 0 mod 2. Similarly, w_{2} = 0 mod 2, so that the V–cobordism X_{0} is spin.

Let X_{0} be a V–cobordism represented by a decorated plumbing diagram Γ, and v_{j} a vertex outside of all the decorating circles. We say that v_{j} is

18 NIKOLAI SAVELIEV

in the Wu set of X_{0} if w_{j} = 1 mod 2. Thus X_{0} is spin if and only if its Wu set is empty. Proposition 6 implies that the Wu set of X_{0} is calculated by the same rules as the Wu set of (undecorated) graph Γ, see Section 2.5. Therefore, one can simply say that the Wu set of X_{0} consists of the vertices of Γ which belong to the Wu set of Γ and sit outside all of the decorating circles. This observation implies the following result.

Corollary. Every plumbed homology sphere bounds a spin plumbed V–cobordism.

Proof. Let Σ be an integral homology 3-sphere obtained by plumbing according to a graph Γ. Since the Wu set of Γ contains no adjacent vertices, we can draw circles around all vertices v_{i} with w_{i} = 1 so that no edge will be intersected by two circles. An extra precaution should be taken not to get a zero-weighted vertex inside any of the circles. This can be easily avoided by blowing up the corresponding vertices. The graph Γ with so drawn circles represents a spin plumbed V–cobordism W_{0} with boundary Σ. Note that this of course is not the most economical method of constructing a spin cobordism in terms of its second Betti number – one can do much better by including as many vertices inside each circle as possible making sure to encircle all the vertices in the Wu set.

3.4. Spin V–cobordisms for Seifert fibered homology spheres. Let Σ(a_{1}, …, a_{n}) be a Seifert fibered homology sphere such that one of the invariants, say a_{1}, is even. Using the freedom allowed by the equation (2), one can choose Seifert invariants (b; (a_{1}, b_{1}), …, (a_{n}, b_{n})) so that b_{1} is odd, b is even and all b_{i} with i = 2, …, n, are even. Each a_{i}/b_{i} admits then a continued fraction expansion [t_{i1}, …, t_{im_{i}}] with all t_{ij} even, so that the manifold obtained by plumbing on the graph in Figure 1 is spin. We can make it into a spin V–cobordism X_{0} by decorating as shown in Figure 15.

We see immediately that H_{2}(X_{0}, ℤ) = ℤ. The rational intersection form of X_{0} is b − ∑ b_{i}/a_{i} = −1/a_{1} ⋯ a_{n}, so that it is negative definite, with b_{+} = 0 and b_{−} = 1.

Let Σ(a_{1}, …, a_{n}) be a Seifert fibered homology sphere with all invariants a_{i} odd. For any choice of Seifert invariants (0; (a_{1}, b_{1}), …, (a_{n}, b_{n})), consider the standard plumbing graph Γ as in Figure 1 (we choose b = 0 for the sake of simplicity). The central vertex of Γ weighted by 0 is in the Wu set for

PSfrag replacements HOMOLOGY 3–SPHERES 19

t_{11}

t_{1m_{1}}
t_{21}
t_{2m_{2}}
t_{n1}
t_{nm_{n}}
b

Figure 15

any choice of Seifert invariants. Blow Γ up (n − 1) times as shown in Figure 16. The 0-framed vertices adjacent to the central vertex are not in the Wu set of the blown up graph, so if we decorate it as shown in Figure 16 we will get a spin plumbed V–cobordism X_{0}.

PSfrag replacements t_{21} t_{2m_{2}} t_{n1} t_{nm_{n}}

t_{11} t_{1m_{1}}

0 0

Figure 16

Proposition 7. The signature of the rational intersection form B of spin plumbed V–cobordism X_{0} shown in Figure 16 equals ∑_{i=1}^{n} sign(a_{i}/b_{i}) − 1.

Proof. Let us consider the decorated plumbing graph in Figure 17 and the associated rational intersection form B′. The (undecorated) plumbing graphs in Figures 16 and 17 represent the same plumbed homology sphere, and one can check that their intersection forms have equal signatures. Splitting these forms over the rationals, we get

∑_{i=1}^{n} sign Δ(0, t_{i1}, …, t_{im_{i}}) + sign B = ∑_{i=1}^{n} sign Δ(t_{i1}, …, t_{im_{i}}) + sign B′.

Since sign Δ(t_{i1}, …, t_{im_{i}}) = sign Δ(0, t_{i1}, …, t_{im_{i}}) + sign(a_{i}/b_{i}) by the diagonalization algorithm, and sign B′ = −1, we are finished.

20 NIKOLAI SAVELIEV

PSfrag replacements t_{21} t_{2m_{2}} t_{n1} t_{nm_{n}} t_{11} t_{1m_{1}} 0

Figure 17

3.5. Splicing plumbed V–cobordisms. Given two plumbed V–cobordisms described by decorated plumbing graphs, we splice them as (undecorated) plumbing graphs as shown in Figure 10, and keep the decorations on both components unchanged. Thus in the gluing region the spliced graph may look as shown in Figure 18.

PSfrag replacements

a
a′
e_{n}
e′_{m}

Figure 18

Note that the two new vertices, framed in Figure 18 by a and a′, do not belong to the Wu set of the spliced graph, as by removing either of them together with the two adjacent edges, we split the graph in two connected components with determinants 0 and ±1. The latter follows easily from the formulas for a and a′ given in Section 2.4. Therefore, the splice of two spin plumbed V–cobordisms is again a spin plumbed V–cobordism.

4. The w–invariants for spin plumbed V–cobordisms

An explicit formula for the w–invariant given in [6] is used to prove that, for any spin plumbed V–manifold (X, c) with boundary Σ, we have w(Σ, X, c) = −μ̄(Σ). This implies in particular that given a plumbed integral homology sphere Σ, the integer w(Σ, X, c) is independent of the choice of a spin plumbed V–cobordism (X, c).

4.1. A formula for the w–invariants. Let X be a compact oriented spin V–manifold with b_{1}(X) = 0, ∂X = Σ, and with only isolated singularities

HOMOLOGY 3–SPHERES 21

which are the cones over lens spaces, L(p_{i}, q_{i}). Then ind D(X ∪ Y) and therefore w(Σ, X, c) can be calculated explicitly by applying the V–index theorem. Such a calculation in [6] provides us with the formula

w(Σ, X, c) = −\frac{1}{8} (sign(X) + ∑ σ(q_{i}, p_{i}, ε_{i})) ,  (7)

where sign(X) stands for the signature of the rational intersection form of X, and functions σ(q, p, ε) for any relatively prime integers p and q and ε = ±1 are defined by the formula

σ(q, p, ε) = \frac{1}{p} ∑_{k=1}^{|p|−1} (cot (\frac{πk}{p} ) cot (\frac{πkq}{p} ) + 2ε^{k} csc (\frac{πk}{p} ) csc (\frac{πkq}{p} )) .

For a spin plumbed V–manifold X, the parameter ε can be calculated as follows. Let X be described by a decorated plumbing graph Γ and let L(p, q) be the lens space obtained by plumbing on a chain Γ′ inside one of the decorating circles. Suppose that the chain looks like the one in Figure 2, with −p/q = [t_{1}, …, t_{n}]. The Wu set of the graph Γ is uniquely determined; in particular, we know which of the vertices of Γ′ belong to this Wu set. We set ε = 1 if the vertex of Γ′ weighted by t_{1} is in the Wu set, and ε = −1 otherwise. In particular, if p is odd, ε = (−1)^{q−1}.

The paper [6] gives us the formula ε = (−1)^{w(m)−1} where L(p, q) is thought of as (−p/q)-surgery on an unknot K in S^{3} with meridian m, and with the spin structure given by w ∈ H^{1}(S^{3} \ K, ℤ/2). The latter obviously agrees with the spin structure described by plumbing on a chain, so the consistency of two formulas for ε will follow from the next lemma.

Lemma 8. The intersection of the Wu set of Γ with the set of vertices of Γ′ is a Wu set of Γ′.

Proof. If p is odd then L(p, q) is a ℤ/2–homology sphere. Therefore, the intersection form A(Γ) of P (Γ) splits over ℤ/2 in two summands, one of which is A(Γ′), and hence the Wu set also splits. If p is even, blow up Γ near Γ′ as shown in Figure 19.

PSfrag replacements t_{1} t_{2} t_{n} 0

22 NIKOLAI SAVELIEV

Let Γ′′ be the chain with vertices framed by 0, t_{1}, …, t_{n}. Since det A(Γ′′) is odd we can apply the argument above and conclude that the restriction of the Wu set of Γ on Γ′′ is a Wu set of Γ′′. To complete the proof we only need to check that the restriction of the Wu set of Γ′′ on Γ′ is a Wu set of Γ′, which is immediate.

4.2. The σ–function. The function σ(q, p, ε) has the following properties, proved in [6], which in practice allow to calculate it without actually referring to its definition :

(1) σ(q + cp, p, ε) = σ(q, p, (−1)^{c}ε);

(2) σ(−q, p, ε) = σ(q, −p, ε) = −σ(q, p, ε);

(3) If p + q is odd then σ(p, q, −1) + σ(q, p, −1) = sign(pq), in particular, σ(1, p, −1) = − sign p for any even p;

(4) For any p, q such that |p| > |q| and p+q is odd, there exists a continued fraction decomposition p/q = [t_{1}, …, t_{n}] with all t_{i} even and |t_{i}| ≥ 2. Then σ(q, p, −1) = − ∑_{i=1}^{n} sign t_{i}.

We will show that in fact σ(q, p, ε) is nothing else but the μ̄–invariant of L(p, q). To be more precise, if p is odd then L(p, q) is a ℤ/2–homology sphere, so the μ̄–invariant is defined as in Section 2.5 and only depends on L(p, q) and not on the choice of a plumbing graph. The only difference is that μ̄(L(p, q)) is not an integer although 8·μ̄(L(p, q) still is. If p is even, there are two different Wu classes because of the fact that H^{1}(L(p, q), ℤ/2) = ℤ/2. Suppose that L(p, q) is obtained by plumbing on a chain Γ′ like the one shown in Figure 2 so that −p/q = [t_{1}, …, t_{n}]. Since q = ±[t_{2}, …, t_{n}] is odd, the Wu set of A(Γ′) is uniquely determined by whether the first vertex of Γ′, weighted by t_{1}, is in the Wu set. We say that v = 1 if it is in the Wu set and v = 0 otherwise. Fixing v allows to define the μ̄–invariants μ̄(L(p, q), v) as before.

Proposition 9. If p is odd then σ(q, p, (−1)^{q−1}) = 8 μ̄(L(p, q)). If p is even, σ(q, p, (−1)^{v−1}) = 8 μ̄(L(p, q), v).

Proof. First we note that we may assume without loss of generality that |p| > |q|. Suppose that p is odd and q is even. We find a continued fraction decomposition p/q = [t_{1}, …, t_{n}] as in Property (4) and evaluate σ(q, p, −1) = − ∑_{i=1}^{n} sign t_{i} = − sign Δ(t_{1}, …, t_{n}) (the latter equality is

HOMOLOGY 3–SPHERES 23

true because all |t_{i}| are greater than or equal to 2). Since the Wu set is empty, 8 μ̄(L(p, q)) = − sign Δ(t_{1}, …, t_{n}).

If both p and q are odd, we use Property (1) to get σ(q, p, 1) = σ(q − p, p, −1) with even q − p. By the previous argument, σ(q − p, p, −1) = 8 μ̄(L(p, q − p)). Let −p/(q − p) = [t_{1}, …, t_{n}] with all t_{i} even, then −p/q = [0, 1, t_{1}, …, t_{n}] with Wu set consisting of just one vertex weighted by 0. Since sign Δ(t_{1}, …, t_{n}) = sign Δ(0, 1, t_{1}, …, t_{n}) by the diagonalization algorithm, we see that μ̄(L(p, q − p)) = μ̄(L(p, q)), which completes the proof in this case.

If p is even, one should consider two separate cases, v = 0 and v = 1. The first is treated as the case of odd p and even q, with all even t_{i} which provides for v = 0. The case v = 1 is completely similar to the case of odd p and q, keeping in mind that adding two vertices framed by 0 and 1 does not change the Wu set of Δ(t_{1}, …, t_{n}).

It should be mentioned that Proposition 9 may provide for easy proofs of various number-theoretical properties of σ(q, p, ε) proved in [6].

4.3. Proof of Theorem 1. Let X be a spin plumbed V–manifold described by a decorated plumbing graph Γ with chains Γ′_{i}, i = 1, …, n, inside the decorating circles, so that ∂P (Γ) = Σ and ∂P (Γ′_{i}) = L(p_{i}, q_{i}). Let w_{i} be the Wu classes of A(Γ′_{i}), i = 1, …, n, and w the Wu class of A(Γ). Since X is spin, w = w_{1} + … + w_{n}. Then

w(Σ, X, c) = −(sign X_{0} + ∑ σ(q_{i}, p_{i}, ε_{i}))/8,  by formula (7),
= −(sign X_{0} + ∑(sign P (Γ′_{i}) − w_{i} · w_{i}))/8, by Proposition 9,
= −(sign X_{0} + ∑ sign P (Γ′_{i}) − w · w)/8,  since X_{0} is spin,
= −(sign P (Γ) − w · w)/8,  by formula (6),
= −μ̄(Σ).

In particular we see that w(Σ, X, c) is independent of the choice of (X, c) as long as we stay in the class of spin plumbed V–manifolds.

5. Homology cobordism properties of the μ̄–invariant

Using Theorem 1, we can reformulate Theorem 5 of Fukumoto and Furuta [5] as follows.

24 NIKOLAI SAVELIEV

Theorem 10. Let Σ be a plumbed homology sphere bounding a spin plumbed V–manifold X. If Σ has finite order in the homology cobordism group then either μ̄(Σ) = 0 or −b_{−}(X) < −μ̄(Σ) < b_{+}(X).

In this section, we apply this result to study the homology cobordism properties of Seifert fibered and plumbed homology spheres. We generalize results obtained in this direction in [3], [5], and [6].

5.1. Proof of Theorem 2. Let Σ = Σ(a_{1}, …, a_{n}) be a Seifert fibered homology sphere with one of the a_{i} even. Let X be the plumbed V–cobordism shown in Figure 15. It is spin and it has b_{−}(X) = 1 and b_{+}(X) = 0. By Theorem 10, if Σ has finite order in the homology cobordism group, μ̄(Σ) must vanish.

Let Σ = Σ(a_{1}, …, a_{n}) be a Seifert fibered homology sphere with all the a_{i} odd, and suppose it has finite order in the homology cobordism group. Choose invariants b_{i} so that all a_{i}/b_{i} but one are positive. Then the plumbed spin V–cobordism X shown in Figure 16 has b_{+}(X) + b_{−}(X) = n − 1 and b_{+}(X) − b_{−}(X) = n − 3, see Proposition 7. Therefore, b_{+}(X) = n − 2 and b_{−}(X) = 1, and Theorem 10 implies that

0 ≤ −μ̄(Σ) < n − 2.  (8)

Now choose a different set of invariants (a_{i}, b_{i}) so that all a_{i}/b_{i} but one are negative. The above argument with V–cobordism in Figure 16 shows that

1 − n < −μ̄(Σ) ≤ 0.  (9)

These inequalities together prove that μ̄(Σ) vanishes.

There is another way to prove the theorem for Seifert fibered homology spheres Σ = Σ(a_{1}, …, a_{n}) with all a_{i} odd. It uses inequality (8) and the result of [15] which says that if an algebraic link has finite order in the homology cobordism group, its μ̄–invariant is non-negative. Since Σ(a_{1}, …, a_{n}) is an algebraic link, we conclude that −μ̄(Σ) ≤ 0, which completes the proof.

5.2. Proof of Theorem 3. Let us denote Seifert invariants of Σ(a_{1}, …, a_{n}) and Σ(α_{1}, …, α_{m}) respectively by (a_{i}, b_{i}) and (α_{j}, β_{j}). The splicing construction provides us with the plumbing graph Γ shown in Figure 20, which we decorate as indicated. One may assume without loss of generality that the V–cobordism X we obtain is spin : if all the a_{i} and α_{j} are odd this is

HOMOLOGY 3–SPHERES 25

automatic; if some of them are even certain precautions should be made in choosing b_{i} and β_{j}.

PSfrag replacements t_{11} t_{1n_{1}} τ_{11} τ_{1m_{1}}

u_{1} u_{k} 0

Figure 20

We use here the notations a_{i}/b_{i} = [t_{i1}, …, t_{in_{i}}], α_{j}/β_{j} = [τ_{j1}, …, τ_{jm_{j}}], p/q = [u_{1}, …, u_{k}] and p/q^{∗} = [u_{k}, …, u_{1}], where the parameters p, q and q^{∗} were computed in Section 3.2.

Let us denote the rational intersection form of X by B, and consider another plumbed V–cobordism X′ given by the plumbing graph Γ′ decorated as shown in Figure 21.

PSfrag replacements t_{11} t_{1n_{1}} τ_{11} τ_{1m_{1}} u_{1} u_{k} 0

Figure 21

Let B′ be the rational intersection matrix of X′. One can easily see that the matrices A(Γ) and A(Γ′) have the same signature. Decomposing these two matrices over the rationals provides us with the identity

sign B + ∑_{i=1}^{n−1} sign Δ(0, t_{i1}, …, t_{in_{i}}) + ∑_{j=1}^{m−1} sign Δ(0, τ_{j1}, …, τ_{jm_{j}})

+ sign Δ(0, u_{1}, …, u_{k}, 0) = sign B′ + ∑_{i=1}^{n−1} sign Δ(t_{i1}, …, t_{in_{i}})

+ ∑_{j=1}^{m−1} sign Δ(τ_{j1}, …, τ_{jm_{j}}) + sign Δ(u_{1}, …, u_{k}).

By diagonalization algorithm, sign Δ(0, t_{i1}, …, t_{in_{i}})−sign Δ(t_{i1}, …, t_{in_{i}}) = − sign(a_{i}/b_{i}) and similarly for α_{j}/β_{j}. The difference sign Δ(0, u_{1}, …, u_{k}, 0)

26 NIKOLAI SAVELIEV

− sign Δ(u_{1}, …, u_{k}) is computed as follows. Decompose the integral intersection matrix Δ(0, u_{1}, …, u_{k}, 0) over the rationals in the direct sum of Δ(u_{1}, …, u_{k}) and the rational intersection matrix U of the decorated plumbing graph shown in Figure 22.

PSfrag replacements u_{1} u_{k} 0

Figure 22

Then sign Δ(0, u_{1}, …, u_{k}, 0) − sign Δ(u_{1}, …, u_{k}) = sign U. The two-by-two matrices B′ and U are computed by the usual rules as

B′ = (\begin{matrix}− ∑ b_{i}/a_{i} − q/p & −1/p \\ −1/p & − ∑ β_{j}/α_{j} − q^{∗}/p \end{matrix}) , U = (\begin{matrix}−q/p & −1/p \\ −1/p & −q^{∗}/p \end{matrix}) ,

and the resulting formula for sign B is

sign B = sign B′ − sign U + ∑_{i=1}^{n−1} sign(a_{i}/b_{i}) + ∑_{j=1}^{m−1} sign(α_{j}/β_{j}).

After simplification using explicit formulas from Section 3.2, we obtain

B′ = (\begin{matrix}α_{m}/(a_{1} ⋯ a_{n−1} · p) & −1/p \\ −1/p & a_{n}/(α_{1} ⋯ α_{m−1} · p) \end{matrix})

and det B′ = −1/(a_{1} · a_{n−1} · α_{1} · α_{m−1} · p). Thus sign B′ = 0 if p > 0, and sign B′ = −2 if p < 0 (the latter could be seen alternatively from the fact that, in the case when p is negative, Σ is an algebraic link and therefore bounds a negative definite plumbed manifold). Concerning matrix U, we compute

det U = −p^{∗}/p

= \frac{a_{1} ⋯ a_{n−1} · α_{1} ⋯ α_{m−1} ∑ b_{i}/a_{i} ∑ α_{j}/β_{j} − a_{n} · α_{m}(b_{n}/a_{n})(β_{m}/α_{m})}{a_{1} ⋯ a_{n−1} · α_{1} ⋯ α_{m−1} − a_{n} · α_{m}},

HOMOLOGY 3–SPHERES 27

where the summation extends over i = 1, …, n − 1 and j = 1, …, m − 1. Replacing ∑ b_{i}/a_{i} by ∑ b_{i}/a_{i} + k at the expense of changing b_{n}/a_{n} to b_{n}/a_{n} − k, and doing the same with the other set of invariants, we see that det U is positive at least for large |k|. If k is large positive, sign U = −2; if k is negative and large in absolute value, sign U = 2.

Suppose that Σ has finite order in the homology cobordism group, and that p is positive. Then sign B′ = 0. If we choose b_{i}/a_{i}, i = 1, …, n − 1, and α_{j}/β_{j}, j = 1, …, m − 1, to be large positive then sign B = n + m − 4. Since the rank of B is n + m − 2, we conclude that b_{+} = n + m − 3, b_{−} = 1, and

−1 < −μ̄(Σ) < n + m − 3.

If we choose b_{i}/a_{i}, i = 1, …, n − 1, and α_{j}/β_{j}, j = 1, …, m − 1, to be negative and of large absolute value then sign B = −n − m + 4 so that b_{+} = 1, b_{−} = n + m − 3, and

3 − n − m < −μ̄(Σ) < 1.

These two sets of inequalities on μ̄(Σ) imply that μ̄(Σ) = 0.

Suppose now that p is negative then sign B′ = −2 and arguing as above we get inequalities

−2 < −μ̄(Σ) < n + m − 4 and 2 − n − m < −μ̄(Σ) ≤ 0,

which imply that μ̄(Σ) is either zero or one. This completes the proof.

References

[1] D. Eisenbud, W. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Princeton Univ. Press, Princeton, 1985

[2] R. Fintushel, R. Stern, Pseudofree orbifolds, Ann. Math. 122 (1985), 335–364.

[3] Y. Fukumoto, On an invariant of plumbed homology 3–spheres. Preprint, 1999

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn

Department of Mathematics, Tulane University, New Orleans, LA 70118

E-mail address: saveliev@math.tulane.edu