Homology 3-spheres bounding acyclic 4-manifolds

Y. Fukumoto^{∗} M. Furuta ^{†}

April 2, 1998

Abstract

We define an integral lift of μ invariant for some class of oriented homology 3-spheres. This invariant is a homology cobordism invariant. If Σ is a homology 3-sphere in this class and the invariant is not zero for Σ, then the connected sum of any number of the copies of Σ cannot be the boundary of an acyclic 4-manifold. The Seifert fibered homology 3-sphere Σ(a_{1}, a_{2}, …, a_{n}) is in this class if one of the a_{i}’s is even. We use Seiberg-Witten’s monopole equation on four-dimensional V-manifolds to show the property of the invariant.

1 Introduction

In this note we give a necessary condition for some homology 3-spheres to be boundaries of acyclic 4-manifolds. R. Stern [26], S. Akbulut and R. Kirby [2] and A. J. Casson and J. L. Harer [3] gave examples of Seifert fibered homology 3-spheres which are the boundaries of acyclic 4-manifolds.

If the μ invariant of a homology 3-sphere is 1 in Z_{2}, then it cannot be the boundary of an acyclic 4-manifold. The μ-invariant is related to the conjecture that any closed topological manifold of dimension≥ 5 is simplicially triangulable. It is known ([18], [15]) that if a homology 3-sphere has non-vanishing μ-invariant and the connected sum of two copies of it is the boundary of an acyclic 4manifold, then the conjecture holds.

In this note we show:

Theorem 1 Let Σ(a_{1}, a_{2}, …, a_{n}) be a Seifert fibered homology 3-sphere with non-vanishing μ-invariant. If one of the a_{i}’s is even, then every connected sum of any number of its copies cannot be the boundary of an acyclic 4-manifold.

∗Department of Mathematics, Faculty of Science, Kyusyu University

†Max-Planck-Instutut für Mathematik, on leave from RIMS, Kyoto University

For invariants of homology 3-spheres introduced before gauge theory, see [21] and [25]. Some obstructions other than μ-invariant are provided by gauge theory. When a homology 3-sphere is the boundary of a negative definite 4manifold whose intersection form is not standard, then a theorem of Donaldson implies that this homology 3- sphere cannot be the boundary of a positive definite 4-manifold.

More direct argument for Seifert fibered homology 3-spheres was given by R. Fintushel and R. Stern [5]. They used the fact that Seifert fibered homology 3-sphere is the boundary of a negative definite V-manifold. (Here we are fixing a convention of orientation.) Their argument was generalized in [4] [17] [11] [10] and [7]. In this note we shall also use this V-manifold introduced in [5].

Main difficulty comes from non-compactness of the moduli space of instantons. T. Lawson [17] used a partial classification of instantons on L(p, q) × R to control the non-compactness. To measure the non-compactness, Fintushel and Stern [7] and the second author [10] introduced an invariant by using the Chern-Simons invariant of flat connections and they extended the argument to apply to the connected sum of various Seifert fibered homology 3-spheres with both orientations. A. Floer pointed out to the second author that a refinement of this invariant, defined by using a version of Floer homology, is a homology cobordism invariant. As was suggested in [9] for cobordisms among lens spaces, if one makes use of the classification of the instantons on L(p, q) × R obtained in [1] [13], then it might be possible to obtain stronger results. However it was not clear how to handle the complicated non-compactness.

K. A. Froyshov investigated definite 4-manifolds with boundaries by using Seiberg-Witten’s monopole equation [8]. Since the moduli space of monopole on closed 4-manifold is compact, it is technically easier to use monopole rather than instanton. For manifolds with boundary, or open manifolds, the moduli space is still non-compact and Froyshov defined an invariant to measure the non-compactness. T. Mrowka, P. Ozsvath and B. Yu [19] analyzed monopole equation of the product on the Seifert fibered 3-manifolds and R, and calculated Seiberg-Witten-Floer homology for these manifolds.

L. I. Nicolaescu [20] recently calculated this invariant of Froyshov for Seifert fibered homology 3-spheres and showed that they cannot be the boundary of acyclic or definite 4-manifolds of some type, under some condition.

All the above arguments to use gauge theory could be basically regarded as attempts to extend Donaldson’s theorem about definite closed 4-manifold to some theorems for open manifolds or V-manifolds.

On the other hand, Donaldson also showed that when a closed 4-manifold is almost definite and spin, one still has some restriction on its intersection form. It would be natural to try to extend the theorem for open or V-cases. However because of the difficulty of non-compactness, it was hard to realize this idea by using instanton moduli spaces.

In [23] N. Saveliev used a restriction on the intersection form of spin closed 4-manifold, which was proved by using monopole moduli space in [12], to give other examples which cannot be the boundary of acyclic 4-manifolds.

In this note we directly use monopole moduli space to extend the theorem in [12] and to obtain some property of almost definite closed V-manifolds. We do not directly consider monopole equation on 4-manifolds with boundaries. H. Tokui [27] applied a similar argument to 4-manifolds with boundary.

Let a_{1}, a_{2}, …, a_{n} be pairwise-coprime positive integers. Then we have a Seifert fibered homology 3-sphere called Σ(a_{1}, a_{2}, …, a_{n}). We define an integer w(a_{1}, a_{2}, …, a_{n}; m) for each integer m using a formula given in Section 5. This number is essentially equal the degree of some class of Seiberg-Witten-Floer homology group.

One of the theorems we show in this note is:

Theorem 2 1. If the connected sum of some number of copies of Σ(a_{1}, a_{2}, …, a_{n}) is the boundary of an acyclic or positive definite 4-manifold, then for any m we have

w(a_{1}, a_{2}, …, a_{n}; m) ≤ 0.

2. Suppose one of the a_{i}’s is even. If the connected sum of some number of copies of Σ(a_{1}, a_{2}, …, a_{n}) is the boundary of an acyclic 4-manifold, then

w(a_{1}, a_{2}, …, a_{n}; s) = 0,  where  s = \frac{1}{2}\{1 + α(−2 + ∑(1 − \frac{1}{a_{i}}))\}.

When one of the a_{i}’s is even, the integer w(a_{1}, a_{2}, …, a_{n}; s) turns out to be an integral lift of the μ-invariant. Theorem 1 is obtained from this observation and Theorem 2.

Casson’s invariant is one of the integral lifts. However the above number does not coincide with Casson’s invariant in general.

In Section 2, we collect all the inputs that we need later on. These arguments are shown by using monopole equation on 4-dimensional closed V-manifolds. In Section 3, we consider their direct consequences to obtain some properties of V-manifolds with boundaries. In Section 4, we define some classes S(k^{+}, k^{−}) of oriented homology 3-spheres for non-negative integers k^{+} and k^{−}. The integral lift of μ-invariant is constructed on S(k^{+}, k^{−}) when k^{+} and k_{−} satisfy k^{+} + k^{−} ≤ 2. In Section 5, we show that the Seifert fibered homology 3-sphere Σ(a_{1}, a_{2}, …, a_{n}) is in the class S(0, 1) and give some examples.

2 Closed V-4-manifolds

In this note we deal with V-manifold with isolated singular points. For general background of V-manifold, see [22] [14] and [19] For general background of Vmanifold. Basically there is no problem to extend usual notions of differential geometry to the ones of V-manifolds if they are defined in the equivariant setting under the finite group action. Let X be an oriented closed V-4-manifold which only has isolated singular points. The neighborhood of each singular point is isomorphic to the form of the cone of a lens space or other spherical space form. Let b_{i} be the i-th Betti number of X and b_{2}^{+}(X) (and b_{2}^{−}(X)) be the maximal dimension of positive (and negative, respectively) definite subspace of H^{2}(X, R). We assume that

(1) b_{1}(X) = 0 and

(2) a V-spin^{c}-structure on X is given and fixed.

Take a V-Riemannian metric on X. Let S^{+}, S^{−} and L be the associated positive and negative spinor bundles and the associated V-line bundle. Then we have ∧^{2}S^{+} = ∧^{2}S^{−} = L. They are V-bundles over X. ¿From now on we omit mentioning “V” since everything is defined in the category of V-manifolds.

Let D_{ASD}(X) be the elliptic complex

D_{ASD}(X) = d + d^{∗} : Ω^{0}(X) ⊕ Ω_{+}^{2}(X) → Ω^{1}(X)

and D_{Dirac}(X) be the Dirac operator associated with the spin^{c}-structure.

We need the indices of D_{ASD}(X) and D_{Dirac}(X). Here the index is the difference between the real dimension of the kernel and that of the kernel of the operator. Since the kernel and the cokernel of D_{Dirac}(X) have complex structures, ind D _{Dirac} is even. These indices can be calculated by using the index theorem for V- manifold of T. Kawasaki [16]. However the index of D_{ASD} is immediately calculated by using harmonic V-forms and Hodge theory for Vmanifold. Since the de Rham cohomology group of any cone, defined by using V-forms, is isomorphic to that of one point, it is a straightforward way to extend the usual de Rham theorem to the V-manifolds. This V-manifold version of de Rham theorem implies the following lemma.

Lemma 1

ind D_{ASD}(X) = 1 + b_{2}^{+}(X).

Note that we are assuming that b_{1}(X) = 0.

The formal dimension of the monopole moduli space for the spin^{c}-structure is given by ind D_{Dirac}(X) − ind D_{ASD}(X). P. Kronheimer pointed out that for negative definite closed 4-manifolds the formal dimension must be negative. The argument is applicable even for certain non-compact 4-manifolds as long as the monopole moduli space is compact [8]. We use V-manifold version of this statement. Since we are considering closed V-manifolds, the moduli spaces are always compact. Hence we have:

Theorem 3 If b_{2}^{+}(X) = 0, then we have ind D_{Dirac} ≤ 0.

The following theorem is proved in [12] for smooth closed spin 4-manifolds. This proof can immediately be applied to closed spin V-4-manifolds as well.

Theorem 4 Suppose the spin^{c}-structure comes from a spin structure. Then ind D_{Dirac}(X) is divisible by 4. If ind D_{Dirac}(X) is not zero, then we have

1 + \frac{1}{2}ind D_{Dirac}(X) ≤ b_{2}^{+}.

The r.h.s is equal to ind D_{ASD}(X) − 1 from Lemma 1. We used this identification.

When the spin^{c}-structure comes from a spin structure, the kernel and the cokernel of D_{Dirac} have quaternionic structures and hence ind D_{Dirac}(X) is divisible by 4.

Let −X be X with its orientation reversed. The spin structure of X canonically induces a spin structure of −X. The corresponding Dirac operator is the formal adjoint of the original one. Hence we have ind D_{Dirac}(−X) = −ind D_{Dirac}(X) and, under the assumption of the above theorem, we obtain:

Corollary 1 Suppose the spin^{c}-structure comes from a spin structure.

1. If ind D_{Dirac}(X) is not zero, then we have

1 − \frac{1}{2}ind D_{Dirac}(X) ≤ b_{2}^{−}.

2. If b_{2}^{+}(X), b_{2}^{−}(X) ≤ 2, then we have ind D_{Dirac} = 0.

Especially, if X is a rational homology spin 4-sphere, then ind D_{Dirac} must be zero for the spin structure. This is also shown by using Theorem 3 without assuming the spin structure.

3 V-4-manifolds with boundaries

In this section, let X be a compact oriented V-4-manifold with boundary Σ. We assume that X only has isolated singular points in its interior. Suppose Σ is a homology 3-sphere and X has a spin^{c}-structure.

Let Y be a spin 4-manifold with boundary −Σ. Since Σ × (0, 1) admits a single spin^{c}-structure, we have a spin^{c}-structure on X ∪_{Σ} Y by patching the spin^{c}-structures on X and Y. Moreover, since the space automorphisms of the spin structure on Σ × (0, 1) is connected, the way of patching is unique up to homotopy.

We denote the spin^{c}-structure of X as c.

Definition 1

w(Σ, X, c) := \frac{1}{2}D_{Dirac}(X ∪_{Σ} Y) + \frac{1}{8}(b_{2}^{+}(Y) − b_{2}^{−}(Y)).

Note that each term of the r.h.s is an integer.

Lemma 2 The integer w(Σ, X, c) is independent of the choice of Y and its spin structure.

Proof.

This kind of definition is now well known and we only sketch an argument. Consider the elliptic operator 4D_{Dirac} ⊕ D_{sign}, where D_{sign} is the signature operator and the coefficient “4” indicates the direct sum of four copies of D_{Dirac}. The Atiyah-Singer index theorem implies that the index of this operator vanishes for closed 4-manifolds. Let Y′ be another spin manifold with boundary −Σ. From an excision argument of the indices of the elliptic operators, we can show that the index for X ∪_{Σ} Y is the sum of the indices of the operators for X ∪_{Σ} Y′ and (−Y′) ∪_{Σ} Y. (One way to show this equality directly is to use Kawasaki’s index theorem for V-manifolds.) This implies that the indices for X ∪_{Σ} Y and X ∪_{Σ} Y′ are the same. Then the lemma immediately follows.

One way to make the excision argument work is to use the framework of Atiyah-Patodi-Singer and to consider operators on V-manifolds with boundaries. Another approach could be possible by using pseudo-differential operator on Vmanifolds. The most direct approach would be to use heat kernel method to calculate the indices for X ∪_{Σ} Y and X ∪_{Σ} Y′ to see that the only contribution comes from the singular points.

The same excision or localization argument also implies the following additivity.

Lemma 3

w(Σ_{0}♯Σ_{1}, X_{0}♮X_{1}, c_{0}♮c_{1}) = w(Σ_{0}, X_{0}, c_{0}) + w(Σ_{1}, X_{1}, c_{1}).

Here Σ_{0}♯Σ_{1} is the connected sum of Σ_{0} and Σ_{1}, and X_{0}♮X_{1} is the boundary connected sum of X_{0} and X_{1}.

For i = 1, 2, …, m, let X_{i} be a compact oriented V-4-manifold with boundary Σ_{i} and which only has isolated singular points in its interior. Suppose Σ_{i} is a homology 3-sphere and X_{i} has a spin^{c}-structure c_{i}.

¿From Theorem 3 and Lemma 3 we obtain:

Theorem 5 Suppose X_{1}, X_{2}, …, X_{m} are all negative definite. If the connected sum Σ_{1}♯Σ_{2}♯ … ♯ Sigma_{m} is the boundary of an acyclic 4-manifold, then we have

∑_{i=1}^{m} w(Σ_{i}, X_{i}, c_{i}) ≤ 0

Proof.

Let Y be the acyclic 4-manifold, then apply Theorem 3 to

(X_{1}♮X_{2}♮ … ♮X_{m}) ∪_{Σ_{1}♯Σ_{2}♯...♯Σ_{m}} (−Y).

Now we consider spin structures.

Lemma 4 If c is a spin structure of X, then we have

μ(Σ) = w(Σ, X, c) mod 2.

Especially, if Σ is the boundary of an acyclic 4-manifold, then w(Σ, X, c) must be even.

Proof.

Since μ(Σ) = (b_{2}^{+}(Y) − b_{2}^{+}(Y))/8 mod 2, it suffices to show that the index of D_{Dirac} is divisible by 4. This follows from the quaternionic structure of the spinor bundles associated to the spin structure.

¿From Theorem 4 and Lemma 3 we similarly obtain:

Theorem 6 Suppose all the c_{i}’s are spin structures. If the connected sum Σ_{1}♯Σ_{2}♯ … ♯ Sigma_{m} is the boundary of an acyclic 4-manifold, then we have the following properties of w(Σ_{i}, X_{i}, c_{i}).

1. The sum ∑_{i=1}^{m} w(Σ_{i}, X_{i}, c_{i}) is even.

2. If the above sum is not zero, then we have

1 − ∑_{i=1}^{m} b_{2}^{−}(X_{i}) ≤ ∑_{i=1}^{m} w(Σ_{i}, X_{i}, c_{i}) ≤ −1 + ∑_{i=1}^{m} b_{2}^{+}(X_{i}).

Corollary 2 Suppose c is spin.

1. If b_{2}^{+}(X), b_{2}^{−}(X) is equal to or less than 2 and the connected sum of some odd number of copies of Σ is the boundary of an acyclic 4-manifold, then we have w(Σ, X, c) = 0.

2. If b_{2}^{+}(X), b_{2}^{−}(X) is equal to or less than 1 and the connected sum of some number of copies of Σ is the boundary of an acyclic 4-manifold, then we have w(Σ, X, c) = 0.

4 Invariant of homology 3-spheres

It is convenient to introduce the following notation.

Definition 2 1. Let X be the set of isomorphism classes of triples (Σ, X, c) such that

(a) X is a compact oriented spin V-4-manifold with only isolated singularities in its interior,

(b) Σ is the boundary of X and we assume that Σ is a homology 3-sphere and

(c) c is the spin structure of X.

2.

X (k^{+}, k^{−}) = {(Σ, X, c) ∈ X | b_{2}^{+} ≤ k^{+}, b_{2}^{−} ≤ k^{−}}.

3. S(k^{+}, k^{−}) is the set of the isomorphism classes of oriented homology 3sphere Σ such that (Σ, X, c) is in the class X (k^{+}, k^{−}) for some X and c.

Note that connected sum induces a map

S(k_{0}^{+}, k_{0}^{−}) × S(k_{1}^{+}, k_{1}^{−}) → S(k_{0}^{+} + k_{1}^{+}, k_{0}^{−} + k_{1}^{−}).

The next theorem is a corollary of Lemma 4 and Corollary 2.

Theorem 7 Suppose Σ is in S(1, 1). If μ(Σ) ≡ 1, then the connected sum of any number of copies of Σ cannot be the boundary of an acyclic 4-manifold.

Now we define an integral lift of μ-invariant for some classes including S(1, 1). The invariant w is defined on X.

Theorem 8 Suppose (k^{+}, k^{−}) is equal to (2, 0), (1, 1) or (0, 2), and two oriented homology 3-spheres Σ_{0} and Σ_{1} in the class S(k^{+}, k^{−}) are homology cobordant to each other. By definition there are some triples (Σ_{0}, X_{0}, c_{0}) and (Σ_{1}, X_{1}, c_{1}) in X (k^{+}, k^{−}). Then we have

w(Σ_{0}, X_{0}, c_{0}) = w(Σ_{1}, X_{1}, c_{1}).

Proof.Recall that Σ_{0} and Σ_{1} are homology cobordant to each other if and only if the connected sum of Σ_{0} and −Σ_{1} is the boundary of an acyclic 4manifold. Since X_{0}♮ − X_{1} satisfies the assumption of Corollary 2, we obtain

w(Σ_{0}, X_{0}, c_{0}) + w(−Σ_{1}, −X_{1}, −c_{1}) = 0.

In particular we have

w(Σ_{1}, X_{1}, c_{1}) + w(−Σ_{1}, −X_{1}, −c_{1}) = 0.

Compare these two to get the required equality.

As an immediate corollary of the above theorem, we obtain:

Theorem 9 We have a well-defined map

w(k^{+}, k^{−}) : S(k^{+}, k^{−}) → Z,

for (k^{+}, k^{−}) satisfying k^{+} + k^{−} = 2. The map w(k^{+}, k^{−}) is a homology cobordism invariant.

Remark. The authors do not know if the invariant is a homology cobordism invariant on the union of the three S(k^{+}, k^{−})’s with k^{+} + k^{−} = 2.

The main theorem of this note is the following theorem, which is just a consequence of the theorems above.

Theorem 10 Suppose Σ is in S(k^{+}, k^{−}) for some (k^{+}, k^{−}) satisfying k^{+}+k^{−} ≤ 2.

1.

μ(Σ) ≡ w(k^{+}, k^{−})(Σ) mod 2.

2. If w(k^{+}, k^{−}) is not zero, then the connected sum of any odd number of Σ cannot be the boundary of an acyclic 4-manifold.

3. If w(k^{+}, k^{−}) is not zero and k^{+} + k^{−} is equal to or less than 1, then the connected sum of any number of Σ cannot be the boundary of an acyclic 4-manifold.

5 Seifert fibered homology 3-spheres

In this section we give some examples.

Seifert fibered homology 3-spheres are classified as follows.

Let Z be a V-Riemann surface whose underlying space is CP^{1} with n marked points for n ≥ 3. Let a_{1}, a_{2}, …, a_{n} be the order of the cyclic isotropy groups at the singular points. Assume that a_{i}’s are pairwise coprime to each other. Then the abelian group consisting of all the isomorphism classes of V-line bundles is an infinite order cyclic group. The first Chern class of one of the two generators is equal to 1/α for α = a_{1}a_{2} … a_{n}. Let L_{0} be a V-line bundle with c_{1}(L_{0}) = −1/α. Then other V-line bundles are of the form L_{0}^{k} for some integer k.

We consider the disk bundle X = D(L_{0}) as an oriented V-4-manifold. When the isotropy representation at each singular point of Z acts freely on the fiber of a V-line bundle except for the origin, the associated V-circle bundle is actually smooth. The V-line bundle L_{0} satisfies this condition. Hence we have a smooth oriented 3-manifold S(L_{0}) = ∂X. We write Σ(a_{1}, a_{2}, …, a_{n}) for this oriented 3-manifold. It is easy to see that this is a homology 3-sphere. It is known that every oriented Seifert fibered homology 3-sphere is of the form ±Σ(a_{1}, a_{2}, …, a_{n}). See [14] for details. The canonical line bundle of the total space of L_{0} is isomorphic to the dual of the pull-back of L_{0} ⊗ T Z. Note that T Z is isomorphic to L_{0}^{k} for some k. Then the canonical bundle is the dual of the pullback of L_{0}^{k+1}.

The total space of L_{0} has a canonical spin^{c}-structure coming from its complex structure. As a subset of L_{0}, we have a canonical spin^{c}-structure of X.

Theorem 11 The V-4-manifold X has a spin structure if and only if one of the a_{i}’s is even.

Proof.

Note that the total space of L_{0} is spin if and only if k is odd and hence L_{0}^{k+1} is a square. A calculation of k follows by looking at the Euler characteristic number of Z [14]:

k = \frac{χ(Z)}{−1/α} = α(−2 + ∑(1 − \frac{1}{a_{i}})).

Then we see the theorem from the above expression of k.

Corollary 3 If one of the a_{i}’s is even, then

Σ(a_{1}, a_{2}, …, a_{n}) ∈ S(0, 1),

hence it lies in S(1, 1) and S(0, 2).

To obtain the spin structure, we have to twist the spinor bundle for the canonical spin^{c}-structure. We need the twisting by the square root of the dual of the canonical bundle, i.e., L_{0}^{(k+1)/2}.

Let c(m) be the spin^{c}-structure defined by the twisting of the canonical spin^{c}-structure by L_{0}^{m}. Then c(0) is the canonical spin^{c}-structure and the spin structure is given by c((k + 1)/2).

Now the invariant w(Σ(a_{1}, a_{2}, …, a_{n}), X, c(m)) is calculated by using Vversion of the Atiyah-Singer index theorem [16]. We shall give detailed arguments and also comparison with other invariants elsewhere. In this note we only state the result of calculation.

Theorem 12

w(Σ(a_{1}, a_{2}, …, a_{n}), X, c(m))

= \frac{1}{8} [−\frac{1}{α} \{(2 − ∑_{i=1}^{n}(1 − \frac{1}{a_{i}})) α + 2m − 1\}^{2} + 1

− ∑_{i=1}^{n} \frac{1}{a_{i}} ∑_{l=1}^{a_{i}−1} \{cot (\frac{πl}{a_{i}} ) cot (\frac{πb_{i}l}{a_{i}} )

+ 2 cos (\frac{π(1 + (−2m + 1)b_{i})l}{a_{i}} ) cosec (\frac{πl}{a_{i}} ) cosec (\frac{πb_{i}l}{a_{i}} )\}]

where

b_{i} \frac{α}{a_{i}} ≡ −1 (mod a_{i}), 0 ≤ b_{i} ≤ a_{i} − 1, b_{i} ∈ Z.

In Introduction we used the notation w(a_{1}, a_{2}, …, a_{n}; m) to denote this number.

Theorem 13 Suppose one of the a_{i}’s is even. Then we have

w(Σ(a_{1}, a_{2}, …, a_{n})) = w(a_{1}, a_{2}, …, a_{n}; s),

for

s = \frac{1}{2} \{1 + α (−2 + ∑_{i=1}^{n}(1 − \frac{1}{a_{i}}))\} .

Here w = w(1, 1) or w(0, 2).

More explicitly, we can write:

w(Σ(a_{1}, a_{2}, …, a_{n}))

= \frac{1}{8} [1 − ∑_{i=1}^{n} \frac{1}{a_{i}} ∑_{l=1}^{a_{i}−1} \{cot (\frac{πl}{a_{i}} ) cot (\frac{πb_{i}l}{a_{i}} )

+ 2 cos (\frac{π(1 + (−2s + 1)b_{i})l}{a_{i}} ) cosec (\frac{πl}{a_{i}} ) cosec (\frac{πb_{i}l}{a_{i}} )\}] .

Examples We give some examples of Brieskorn homology 3-spheres such that the con-

nected sum of any number of the copies cannot be the boundary of an acyclic 4-manifold. The following is a list of Brieskorn homology 3-spheres for which Theorem 5 is not applied, but Theorem 6 can be applied.

Brieskorn max{ w }  spin case  Casson’s invariant
Σ(2, 3, 7)  0  w(1) = −1  −1
Σ(4, 5, 7)  0  w(29) = −1  −5
Σ(7, 9, 10)  0  w(204) = −1  −25
Σ(2, 5, 11)  0  w(12) = −1  −3
Σ(5, 8, 11)  0  w(129) = −1  −17
Σ(4, 9, 11)  0  w(109) = −1  −15
Σ(5, 7, 12)  0  w(121) = −2  −16
Σ(7, 11, 12)  0  w(316) = −1  −37
Σ(3, 4, 13)  0  w(27) = −1  −5
Σ(5, 6, 13)  0  w(109) = −1  −15
Σ(3, 8, 13)  0  w(73) = −1  −11

In the above list, only Σ(5, 7, 12) has trivial μ-invariant. Remark.

1. There are other examples in S(k^{+}, k^{−}) (k^{+} + k^{−} = 2). For i = 1, 2, …, m suppose L_{i} is a V-line bundle over a V-Riemann surface Z_{i}. Take any tree graph with m vertices. Write v_{i} (i = 1, 2, …, m) for the vertices. Then one can construct a plumbing of L_{i}’s corresponding to this tree graph. Truncate this V-4-manifold to get a V-4-manifold with boundary. If this boundary is a smooth homology 3-sphere, then it lies in the class S(k^{+}, k^{−}) for some k^{+} and k^{−} satisfying k^{+} + k^{−} = m.

2. The class S(k^{+}, k^{−}) (k^{+} + k^{−} = 2) is defined by oriented homology 3spheres which are the boundaries of almost definite spin V-manifolds. In other words, we used cobordism between the homology 3-sphere and spherical space forms. It is possible to extend the argument by replacing spherical space forms with oriented closed rational homology 3-spheres which admit positive scalar curvature metrics. Moreover when they do not admit positive scalar curvature metric, presumably certain condition about Seiberg-Witten Floer homology could be used instead. By finding this condition, it might be possible to generalize the theorems in this note.

Acknowledgments: The authors is grateful to Masaaki Ue and Vyacheslav S. Krushkal for pointing out some mistakes in the earlier version of the paper. The second author would like to thank the Max-Planck-Institut für Mathematik in Bonn for its hospitality.

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