75_1-3_151

Acta Applicandae Mathematicae 75: 151–165, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

151

Equivariant Casson Invariants andLinks of Singularities

OLIVIER COLLIN1,∗ and NIKOLAI SAVELIEV2

1Département de mathématiques, Université du Québec à Montréal, Montréal,Québec H3C 3P8, Canada. e-mail: [email protected] of Mathematics, Duke University, Durham, NC 27708, U.S.A.e-mail: [email protected]

(Received: 4 April 2002)

Abstract. This article is a survey of the authors’ works over the last few years in trying to understandbetter various problems about the Casson invariant and Floer homology of links of singularities and,more generally, of graph homology spheres.

Mathematics Subject Classifications (2000): 57M12, 57R57, 32S55.

Key words: gauge theory, Casson invariant, Floer homology, links of complex singularities.

Let � be an integral homology 3-sphere with a finite cyclic group action makingit into a branched cover of another integral homology sphere with branch set aknot k. The equivariant Casson invariant of � is defined in [5] by counting SU(2)-representations of π1� which are equivariant with respect to the induced action.Our definition of the equivariant Casson invariant is gauge theoretical (Cappell, Leeand Miller in an unpublished manuscript utilized an intersection theory to define anequivariant Casson invariant for integral homology spheres acted upon by a cyclicgroup of odd prime order). This invariant can be identified with certain equivariantknot signatures of k. Application of this theory to various natural cyclic groupactions on the links of singularities leads to several interesting results, among whichare a geometric proof of the Neumann–Wahl type formulae and a closed formformula for the Floer homology of Seifert fibered homology spheres.

1. The Equivariant Casson Invariant

In this section, after briefly recalling Taubes’ gauge-theoretic construction of theCasson invariant, we define the equivariant Casson invariant and relate it to theTristram–Levine equivariant knot signatures.

* The first author was supported by an NSERC Post-doctoral Fellowship and Harvard University.The work of the second author was partially supported by NSF Grant DMS 0196523.

152 OLIVIER COLLIN AND NIKOLAI SAVELIEV

1.1. THE CASSON INVARIANT

A closed oriented 3-manifold � is called an integral homology sphere if it has thesame integral homology as the 3-sphere. Associated with every integral homologysphere � is an integer λ(�) called the Casson invariant of �. It was defined byCasson around 1985 via a ‘creative count’ of irreducible SU(2) representations ofthe fundamental group of �.

Given an integral homology sphere � let

R(�) = Hom∗(π1(�),SU(2))/ ad SU(2)

be the space of irreducible SU(2)-representations of its fundamental group moduloconjugation. The space R(�) is said to be nondegenerate if for every α ∈ R(�)

one has H 1(π1�, adα) = 0. If R(�) is nondegenerate, it is necessarily finite, andthe Casson invariant λ(�) is defined via a signed count of points in R(�),

λ(�) = 1

2

∑α∈R(�)

εα, εα = ±1.

For the sake of simplicity, we will restrict ourselves to this definition and onlymention that, if R(�) is degenerate, it needs to be perturbed first to make it finiteand then λ(�) can be defined in a similar fashion, see [28].

There exist several ways to describe the integers εα = ±1. In the originaldefinition, they appeared from an intersection theory associated with a Heegaardsplitting of �. Another approach, which is due to Taubes [28], utilizes gauge theoryon � to first define the so called Floer index µ(α) and then let εα = (−1)µ(α). Inthis approach, µ(α) is actually defined modulo 8, and Taubes’ construction canbe refined to define a Z/8-graded homology theory whose Euler characteristic is(twice) the Casson invariant. This theory is called the (instanton) Floer homology,see [10], and also [3] for a less formal introduction.

In this paper, we will pursue Taubes’ approach in defining εα. In the back-ground of this approach lies the well known correspondence between representa-tions π1(�) → SU(2) and flat connections in principal SU(2) bundles over �,which is established by the holonomy map. Note that, by topological reasons, allSU(2) bundles over � are trivial.

Fix a Riemannian metric on � and think of a representation α as a flat con-nection on a trivialized SU(2)-bundle E over �. Let θ be the product connectionand choose a path α(t), 0 � t � 1, of connections on E such that α(0) = θ andα(1) = α. Associated with α(t) is a path of self-adjoint Fredholm operators

Kα(t) =(

0 d∗α(t)

dα(t) − ∗ dα(t)

)(1)

on (�0 ⊕ �1)(�, adE), where adE = E ×ad su(2) is the adjoint bundle of E,and dA is the covariant derivative. The one-parameter family of spectra of operators

EQUIVARIANT CASSON INVARIANTS AND LINKS OF SINGULARITIES 153

Figure 1.

Kα(t) can be viewed as a collection of spectral curves in the plane, see Figure 1,connecting the spectrum of Kθ with that of Kα .

Consider the straight line connecting the points (0,−δ) and (1, δ) where δ > 0is chosen smaller than the absolute value of any nonzero eigenvalue of Kθ and Kα .The spectral flow is the number of eigenvalues, counted with multiplicities, whichcross from ‘below’ to ‘above’ this line minus the number which cross from ‘above’to ‘below’. It is well defined and finite along a generic path.

The sole purpose of the number δ > 0 in the above definition was to make senseof the count near t = 0, where the family Kα(t) has three-dimensional kernel. Thusdefined, the spectral flow is independent modulo 8 of the choices in its definition,see [2], and it is called the Floer index of α.

This completes the definition of the Casson invariant, at least in the nondegener-ate case. With an extra bit of work, one can prove the following properties of λ(�),see [1]. It is invariant with respect to orientation preserving homeomorphisms, itchanges sign when the orientation of � is reversed, it is additive with respect toconnected sums, it changes in a predictable manner with respect to Dehn surgery,and its reduction modulo two is the Rohlin invariant of �. Right from the defin-ition we also see that λ(�) vanishes for simply connected homology spheres �

– of course, the still unresolved Poincaré conjecture asserts that the only simplyconnected homology sphere � is S3.

Remark. It should be noted that although λ(�) is defined via a count of repre-sentations α: π1(�) → SU(2), the signs εα which they are counted with encodesome additional geometric information about �. Therefore, one should not expectλ(�) to be an invariant of π1(�), and in fact it is not. For example, if � is an inte-gral homology sphere such that λ(�) = 0, then manifolds �#� and �#−�, where‘#’ stands for connected sum and ‘−’ for reverse of orientation, have isomorphicfundamental groups but different Casson invariants.

Remark. Finally, we want to clarify why we required that � be an integralhomology sphere. This has to do with reducible SU(2)-representations of its fun-damental group: every such representation is Abelian hence it factors througha representation of H1(�,Z) = 0 and is therefore trivial. Thus we are sparedthe trouble of dealing with nontrivial reducible representations. With an extra ef-fort, see Walker [30], the Casson invariant can be extended to rational homology


spheres, that is, oriented 3-manifolds with the rational homology of S3. The non-trivial reducible representations should be added to the count of the irreducibleones, although with weights in general different from ±1. Lescop [17] gave anextension of the Casson invariant to all oriented 3-manifolds.

1.2. THE EQUIVARIANT CASSON INVARIANT

Let � be an integral homology 3-sphere endowed with an orientation preservingdiffeomorphism τ : � → � of order n. The diffeomorphism τ induces a Z/n-action on � whose quotient is homeomorphic to a rational homology sphere, whichwe denote by �′. From this point on we will assume that the Z/n-action is not freeso that �′ is an integral homology sphere. This condition ensures that �′ has adistinguished branch set, a knot K, corresponding to the nonempty fixed point setof τ .

The diffeomorphism τ induces a map τ ∗: R(�) → R(�) on irreducibleSU(2)-representations of π1� with the fixed point set Rτ (�) = Fix(τ ∗). It alsoinduces an automorphism ν of the bundle E consistent with τ ∗: R(�) → R(�).It should be noted that νn need not be the identity in general.

Let us assume that the space Rτ (�) is nondegenerate, that is, the equivariantcohomology groups H 1

τ (�, adα) vanish for all α ∈ Rτ (�). Note that if R(�) isnondegenerate then so is Rτ (�).

Refining the construction of the Floer index, we associate with every represen-tation α ∈ Rτ (�) an equivariant Floer index as follows. Let us fix a Riemannianmetric on � which is invariant with respect to τ . Choose a path α(t), 0 � t � 1,of connections such that α(0) = θ , α(1) = α and ν∗α(t) = α(t) for all t (thelatter can be achieved by averaging). Consider the path of self-adjoint Fredholmoperators Kν

α(t) on (�0 ⊕ �1)ν(�, adE) obtained by restricting the operators (1)onto the differential forms invariant with respect to the induced action of ν. TheFloer index µτ(α) is then defined to be the spectral flow of Kν

α(t) reduced mod 4(as this spectral flow is only well-defined modulo 4).

If Rτ (�) is nondegenerate, the equivariant Casson invariant is defined by theformula

λτ (�) = 1/2∑

α∈Rτ (�)

(−1)µτ (α). (2)

If Rτ (�) fails to be nondegenerate, one needs to use some perturbations to producea nondegenerate equivariant representation space. The equivariant Casson invariantis then defined by counting points in this space, see [5] for details.

Remark. Thus two essential new features come into the definition of the equivari-ant Casson invariant. First, only equivariant representations are counted, and sec-ond, they are counted with the signs which, in general, might be different from


those used to define the regular Casson invariant. When n is odd, one has the nicerelation µτ(α) = µ(α)mod 2, see [7]. However, such a formula need not be truefor even n, see, for instance, (6).

The equivariant Casson invariant λτ (�) is an invariant of the equivariant ori-entation preserving diffeomorphism type of (�, τ), it is additive with respect toconnected sums of pairs (�, τ), and its reduction modulo 2 equals the Rohlin in-variant (the latter follows from Theorem 2 below). The behavior of the equivariantCasson with respect to (equivariant) surgery is essentially unknown at the moment.

One of the important results in this theory is the following theorem from [5]expressing λτ (�) in terms of the equivariant knot signatures of the branched set.Recall that, for a knot K in an integral homology sphere, the equivariant knotsignature signβ(k), 0 � β � 1, is defined as the signature of the Hermitian form

(1 − e2πiβ) S + (1 − e−2πiβ) St ,

where S is a Seifert matrix of K and St its transpose. The signature sign1/2(k) isusually denoted sign(k) and is called the knot signature.

THEOREM 1. Let � be an integral homology 3-sphere and τ : � → � an ori-entation preserving diffeomorphism of order n. Suppose that the quotient of � bythe induced Z/n-action is an integral homology sphere �′ with branch set a knotK ⊂ �′. Then

λτ (�) = n · λ(�′) + 1

8·n−1∑k=0

signk/n(K).

Proof. A nondegenerate equivariant representation α: π1� → SU(2) such thatτ ∗α = uαu−1 can be pushed down to the quotient �/τ = �′, away from thebranch set K, to give a representation α′: π1(�

′ \ K) → SU(2) with α′(µ)conjugate to u, where µ is a meridian of K. According to a theorem of Herald[14], certain count of representations α′ with a fixed trace of the meridians givesessentially the sum of an equivariant signature of K and the Casson invariant of �′.Therefore, to prove the theorem one compares the count of representations α usedin the definition of the equivariant Casson invariant with that of [14]. This is donewith the help of the orbifold gauge theory developed in [7]. In the degenerate case,special attention should be paid to matching perturbations in the two theories. Fora complete proof, see [5]. ✷

Theorem 1 can be re-interpreted as follows. Let K be a knot in an integralhomology 3-sphere �′, which is the boundary of a smooth oriented 4-manifold M ′with H1(M

′,Z) = 0. Let � be the n-fold cyclic branched cover of �′ with branchset the knot K. Given a Seifert surface FK of the knot K, let M be the n-fold cyclicbranched cover of M whose branch set is the surface FK with its interior pushedslightly into M. Then M is a smooth four-dimensional manifold with boundary �

and with a natural Z/n-action τ : M → M.


THEOREM 2. Suppose that the manifold � = ∂M constructed above from a knotK ⊂ �′ = ∂M ′ is an integral homology sphere. Then

λτ (�) − n · λ(�′) = 18 (sign(M) − n · sign(M ′)).

Proof. This follows easily from Theorem 1 and Viro’s theorem [29] relatingknot signatures with the signatures of branched covers. ✷

Theorem 2 can be refined in the following way. Each of the n equivariant knotsignatures in Theorem 1 can be identified with a corresponding equivariant signa-ture of M, and the equivariant Casson invariant can be expressed as a sum of refinedinvariants taking into account further structure on the equivariant representationspace of �, see [5].

1.3. CASSON VS EQUIVARIANT CASSON

The relation between the Casson invariant and its equivariant counterpart has onlybeen studied in some instances. In particular, the cases of two-fold branched coversof knots in S3, cyclic branched covers of untwisted Whitehead doubles of knots inS3 and also cyclic branched covers of graph knots are all well understood.

EXAMPLE. Let K be a knot in S3 and � a double branched covering of S3 withbranch set a knot K. If � is an integral homology sphere then

λτ (�) − λ(�) = 1

12

d

dt

∣∣∣∣t=−1

lnVk(t),

where Vk is the Jones polynomial of the knot k. This follows immediately fromTheorem 1 and Mullins’ formula [18] for the Casson invariant of double branchedcovers of knots in S3. This should be seen as a first step in expressing the differencebetween the Casson and equivariant Casson invariants in terms of the topology ofthe fixed-point set of the cyclic action.

EXAMPLE. Let K be a knot in S3, and denote by DεK, ε = ±1, its untwistedWhitehead double with ε-clasp. The knot D1K is obtained by embedding the circlein a solid torus as shown in Figure 2 on the left, and then embedding that solidtorus in another copy of S3 as a tubular neighborhood of K with the longitude ofthe solid torus going to the canonical longitude of K, preserving orientations. The(−1)-clasp is shown on the right in Figure 2.

Let � be the n-fold cyclic branched cover of S3 with branch set DεK. Then� is an integral homology sphere and λτ (�) = 0 for any knot K because allthe equivariant signatures of DεK are equal to zero. On the other hand, λ(�) =nε · $′′

K(1) according to [15]. Here, $K(t) is the Alexander polynomial of K

normalized so that $K(1) = 1 and $K(t) = $K(t−1).


Figure 2.

From the two constructions described above, it is not difficult to extract plentyof examples of integral homology spheres � with Z/n-actions such that λ(�) =λτ (�). Very recent work of Garoufalidis and Kricker [13] gives new input into theproblem of measuring the difference between λ(�) and λτ (�) where � is an n-foldcyclic branched cover of S3 with branch set a knot K, and τ is the natural cyclicaction. They give a formula for the difference in terms of residues of a rationalfunction related to the Kontsevitch integral of K.

On the other hand, one may focus on the cases when the two invariants actu-ally agree. There exists a large class of integral homology spheres for which wecan prove that the two invariants coincide. This class consists of cyclic branchedcovers of graph knots, and plays a prominent role in the applications to the links ofsingularities. We describe this class next, following the book [8].

For the purposes of this description, by a link L = (�,L) = (�,K1 ∪ · · · ∪Km) we mean a pair consisting of an oriented integral homology sphere � and acollection L of disjoint oriented knots K1, . . . , Km in �. Given two links, L =(�,L) and L′ = (�′, L′), choose components K ⊂ L and K ′ ⊂ L′ and letN(K) and N(K ′) be their tubular neighborhoods. Denote by m, ( ⊂ ∂N(K) andm′, (′ ⊂ ∂N(K ′) canonical meridians and longitudes. Form

� = (� \ IntN(K)) ∪ (�′ \ IntN(K ′))

pasting along boundaries by matching m to (′ and ( to m′. This operation results ina new link called the splice of L and L′ along K and K ′.

A link (�(a1, . . . , an),K1 ∪ · · · ∪Km) where K1, . . . , Km are fibers in a Seifertfibered homology sphere �(a1, . . . , an), compare with (3), is called a Seifert link.Note that a general Seifert link may include singular fibers as well as some nonsin-gular fibers. Any link which can be obtained from a finite number of Seifert linksby splicing is called a graph link. Empty graph links are called graph homologyspheres. Graph links with one component are called graph knots. Torus knots andall their cables are graph knots in S3.

Remark. The splitting theorem of Jaco-Shalen and Johannson implies that anyirreducible link L = (�,K) can be expressed as a result of splicing together acollection of Seifert links and the so-called simple links (�,L) whose comple-ments � \ IntN(L) admit a complete hyperbolic structure of finite volume (except


possibly if L is empty or � is not sufficiently large). Thus graph links are preciselythe links that admit splicing decomposition without simple components.

THEOREM 3. Let K ⊂ �′ be a graph knot and � be the n-fold cyclic branchedcover of �′ with branch set the knot K. If � is an integral homology sphere thenλτ (�) = λ(�).

This theorem is proved in [5]. It should be mentioned that an explicit formulafor λ(�) where � is an n-fold branched cover over a graph knot was given byNémethi [19]. It is interesting to compare his formula with our formula for λτ (�)

keeping in mind Theorem 3.

2. Links of Singularities and the Neumann–Wahl Formula

In this section, we apply the equivariant Casson theory to the study of the linksof singularities. The main object of our study is the Neumann–Wahl conjectureexpressing certain analytic invariants of a singularity in terms of its topology.

2.1. LINKS OF SINGULARITIES

Given analytic functions f1, . . . , fm: (Cn, 0) → (C, 0) let X = { f1 = 0, . . . , fm =0 } be such that the matrix

J (p) =(∂fi

∂zj

)(p)

has rank n − 2 at all points p ∈ X except perhaps at 0. If rank of J (0) is alsoequal to n − 2, we say that (X, 0) is smooth, that is, analytically isomorphic to(C2, 0). Otherwise, (X, 0) is called an isolated surface singularity. We will assumethat (X, 0) is irreducible and normal. For example, if f : (C3, 0) → (C, 0) satisfiesJ (p) = 0 for any p in a neighborhood of the origin, then (X, 0) = ({ f = 0 }, 0)is a normal surface singularity.

Fix an arbitrary embedding (X, 0) into (Cn, 0). Then there exists ε0 such thatfor any 0 < ε � ε0 the (2n − 1)-dimensional sphere S2n−1

ε = { z ∈ Cn | |z| = ε }

intersects (X, 0) transversally. In particular, for such ε, the intersection

�X = X ∩ S2n−1ε

is a closed three-dimensional real manifold which does not depend on the em-bedding (X, 0) ⊂ (Cn, 0) and on ε. It is canonically oriented as the boundary ofa (singular) complex surface. The manifold �X is called the link of singularity(X, 0).

One of the main questions in the singularity theory is whether one can recoveranalytic invariants of a singularity (X, 0), such as geometric genus pg, from topo-logical information, that is, from the link �X . In general the answer is negative: forexample, the links of singularities at zero of

X1 = {x2 + y3 + z18 = 0} and X2 = {z2 + y(x4 + y6) = 0}


have the same topology but different geometric genus, pg(X1) = 3 and pg(X2) =2. Nevertheless, one may want to find nice families for which the above questioncan be answered in positive. An example of such an approach is the followingNeumann–Wahl conjecture.

2.2. THE NEUMANN–WAHL CONJECTURE

Let � be the link of a normal complete intersection surface singularity and let Mbe the associated Milnor fiber viewed as a compact 4-manifold with boundary �.The signature of M is an analytic invariant which is ‘equivalent’ to the geometricgenus of the singularity. Neumann and Wahl conjectured in [22] that, if � is anintegral homology sphere then λ(�) = 1

8 sign(M).

EXAMPLE. Let p, q, r � 2 be pairwise relatively prime integers then the link ofsingularity at zero of xp +yq +zr = 0 is an integral homology sphere. It is denotedby �(p, q, r) and is usually referred to as a Brieskorn homology sphere. The mapτ : �(p, q, r) → �(p, q, r) given by the formula τ(x, y, z) = (x, y, e2πi/r z) isan orientation preserving diffeomorphism of order r, which makes �(p, q, r) intoan r-fold cyclic branched cover of S3. Its branch set is a (p, q)-torus knot. Theaction τ naturally extends to the Milnor fiber M(p, q, r) of the singularity with thequotient a 4-ball. Therefore,

λ(�(p, q, r)) = λτ (�(p, q, r)) = 18 sign(M(p, q, r)),

where the first equality follows from Theorem 3 and the second from Theorem 2.Hence, the Neumann–Wahl conjecture holds for all Brieskorn homology spheres.For details and ramifications see [6].

EXAMPLE. Given n pairwise relatively prime integers a1, . . . , an � 2, the Seifertfibered homology sphere �(a1, . . . , an) is the link of singularity at zero of thesystem of polynomial equations

bi1za11 + · · · + binz

ann = 0, i = 1, . . . , n − 2, (3)

where bij is a sufficiently general (n−2)×n matrix of complex numbers. If n = 3,one gets back Brieskorn homology spheres. The map

τ(z1, . . . , zn−1, zn) = (z1, . . . , zn−1, e2πi/anzn)

is an orientation preserving diffeomorphism of order an making �(a1, . . . , an)

into a cyclic branched cover of �(a1, . . . , an−1). Its branch set is a graph knot,which is in fact a regular fiber of the Seifert fibration of �(a1, . . . , an−1). The mapτ extends to the Milnor fiber M(a1, . . . , an) so that M(a1, . . . , an) is an an-foldcyclic branched cover of M(a1, . . . , an−1) with branch set a Seifert surface of theknot Kn. By induction on n, we conclude that

λ(�(a1, . . . , an)) = 18 sign M(a1, . . . , an), (4)

hence the Neumann–Wahl conjecture is true for all Seifert fibered homology spheres.


EXAMPLE. Let g(x, y, z) = f (x, y) + zn define an analytic map g: (C3, 0) →(C, 0) with an isolated singularity at 0, and let � be the link of this singularity.The map τ(x, y, z) = (x, y, e2πi/nz) is an orientation preserving diffeomorphismof � of order n, which makes � into a cyclic branched cover of S3 with branch seta graph knot obtained by iterative cabling of the unknot in S3. If � is an integralhomology sphere then λ(�) equals λτ (�) by Theorem 3, and the latter equals oneeighth of the Milnor fiber signature by Theorem 2. This establishes the Neumann–Wahl conjecture for all such links.

The Neumann–Wahl conjecture was verified in the above three special cases byFintushel and Stern [9] and Neumann and Wahl [22] by explicitly computing theboth sides of the formula and identifying them with the help of some combina-torics. Thus far, the main input of the equivariant Casson theory in this area is thatit provides more conceptual proofs. However, it also verifies the Neumann–Wahlconjecture for a wider class of links, namely, links built from S3 by iterating thebranched cover construction, with a graph knot as the branch set at every stage.Among such links are, for instance, the links of singularities given by the systemsof equations of the form fi(z1, . . . , zi+1) + z

nii+2 = 0, i = 1, . . . , k. It should be

mentioned that not all links of singularities can be represented as iterated cyclicbranched covers, and therefore our approach does not prove the Neumann–Wahlconjecture. However, our work suggests conceptual reasons for such a formula tohold or not and also hints that the phenomenon conjectured by Neumann and Wahlgoes beyond Complex Geometry.

Remark. We already mentioned that there exist singularities which cannot becharacterized topologically. In order to avoid such examples, one normally imposescertain analytic and topological restrictions, such as that the link be a rational ho-mology sphere. However, it should be noted that the simple minded extension of theNeumann–Wahl conjecture to rational homology spheres, which replaces the Cas-son invariant with the Casson–Walker invariant, does not hold, see [4]. A versionof Theorem 2 can be worked out in the case of rational homology spheres, usingorbifold representations arising in Floer homology for knots rather than equivariantrepresentations, but this is not satisfactory to give a proper generalization of theNeumann–Wahl conjecture. Another approach to generalizing this conjecture torational homology spheres was recently suggested by Némethi and Nicolaescu [20]using Seiberg–Witten invariants.

3. Special Involutions on Graph Homology Spheres

In the context of finite cyclic group actions on 3-manifolds, links of isolated surfacesingularities often come with an intrinsic involution: the complex conjugation. Inthis case, the equivariant Casson invariant generally does not agree with the Casson


invariant and is therefore of particular interest. We study here the equivariant Cas-son invariant of homology spheres which are more general than the links, namely,graph homology spheres endowed with an involution representing them as doublebranched covers of S3 with branch set a Montesinos knot. We give an explicitformula for the equivariant Casson invariant of such manifolds in terms of the µ̄-invariant of Neumann and Siebenmann and also in terms of the Floer homology.We conclude this section with some topological applications of this invariant.

3.1. THE µ̄-INVARIANT

The basic example for the set up is the following. Let �(a1, . . . , an) be a Seifertfibered homology sphere viewed as the link of singularity (3) with real coefficientsbij . The involution τ(z1, . . . , zn) = (z̄1, . . . , z̄n) induced by complex conjuga-tion represents �(a1, . . . , an) as a double branched cover of S3 with branch seta Montesinos knot.

Given a general graph homology sphere �, one can think of it as being obtainedby splicing Seifert fibered homology spheres together. The complex conjugationinvolution on each of the Seifert fibered pieces yields an involution on � which wecall again τ . This makes � into a double branched cover of S3 with branch set a(generalized) Montesinos knot Kτ . Links of singularities are a special case of thisconstruction.

THEOREM 4. Let � be a graph homology sphere and τ the involution con-structed above. Then λτ (�) only depends on � and not on the choices made inthe definition of τ . Moreover, λτ(�) = µ̄(�), the µ̄-invariant of Neumann andSiebenmann.

Theorem 4 follows immediately from Theorem 2 after we recall that the µ̄-invariant of a graph homology sphere � is defined by the formula µ̄(�) = 1/8 ·sign(Kτ ) and that it only depends on the orientation preserving diffeomorphismtype of �, see [27].

Another definition of the µ̄-invariant is as follows, see [21]. Every graph ho-mology sphere � is the boundary of a plumbed manifold P . For example, if � isa link of singularity then P can be chosen to be a ‘good resolution’ of singularity,that is, one for which the exceptional curve E = ⋃

Ei is a union of smooth curvesintersecting transversally. Then

µ̄(�) = 18(sign P − w · w), (5)

where the dot stands for the intersection product, and w ∈ H2(P,Z) is a homologyclass which is dual to the second Stiefel–Whitney class and each of whose coordi-nates is either 0 or 1 in the canonical basis of H2(P,Z) associated with plumbing(this basis is generated by the Ei if P is a ‘good resolution’).


The µ̄-invariant is easy to compute explicitly. It is additive with respect to splic-ing, so one only needs to compute it for Seifert fibered homology spheres. Explicitformulae for µ̄(�(a1, . . . , an)) in terms of a1, . . . , an can be found in [21].

3.2. FLOER HOMOLOGY OF LINKS OF SINGULARITIES

Floer [10] associated with every integral homology sphere � eight finitely gener-ated Abelian groups Ik(�), k = 0, . . . , 7, called (instanton) Floer homology of �.These groups refine the Casson invariant in that λ(�) equals one half the Eulercharacteristic of the Floer homology of �. The Floer homology is the homologyof the Floer chain complex which is defined as follows. If the representation spaceR(�) is nondegenerate, the generators in the Floer chain complex are points inR(�) graded by their Floer index mod 8. The boundary operator is defined as aproper count of isolated solutions of the anti-self-duality equation on the cylinderR × �, see [3] and [10]. If R(�) is degenerate, perturbations need to be used butthe construction is essentially the same.

As we see, the definition of I∗(�) depends heavily on gauge theory in di-mensions three and four so that these groups in general are difficult to compute.However, the situation is much less complicated with Seifert fibered homologyspheres – their Floer homology groups are completely known. Fintushel and Stern[9] provided an algorithm for calculating Floer homology of Brieskorn homol-ogy spheres, which was later extended to �(a1, . . . , an) in [16]. This algorithmproceeds by explicitly describing generators in the Floer chain complex and com-puting their Floer indices but it becomes increasingly difficult as the n grows.The approach via Theorem 5 below provides a closed form formula for the Floerhomology of �(a1, . . . , an) for arbitrary n. This should be seen as a first step inthe difficult task of understanding the Floer homology of general links of isolatedsurface singularities.

THEOREM 5. If � = �(a1, . . . , an) is a Seifert fibered homology sphere, thenits Floer homology groups are free Abelian. Let bk be the kth Floer Betti number,that is, the rank of the group Ik(�). Then bk = 0 if k is even, and

b1 = b5 = −1/2 · (λ(�) + µ̄(�)),

b3 = b7 = −1/2 · (λ(�) − µ̄(�)).

Proof. Let us first assume that n = 3 so that we deal with a Brieskorn homologysphere � = �(p, q, r). The representation space R(�) is nondegenerate hence itspoints form a basis in the Floer chain complex. According to Fintushel and Stern[9], the Floer index µ(α) is odd for all α ∈ R(�). In particular, this implies thatthe Floer chain complex has trivial boundary operator. The Floer homology groupsof � then coincide with the Floer chain groups and bk = 0 if k is even.

Although the Floer homology is, by definition, graded modulo 8, Frøyshovproved in [11] that the Floer Betti numbers are in fact four-periodic for any ho-mology sphere. In our situation this means that b1 = b5 and b3 = b7. Since the


Casson invariant is one half the Euler characteristic of the Floer homology, weimmediately obtain the relation b1 + b3 = −λ(�).

According to [23], the induced action of the complex conjugation involution τ

on R(�) is trivial, which means that Rτ (�) = R(�). A rather intricate calcula-tion of the equivariant Floer index in [23] shows that

µτ (�) = 12(µ(α) + 1)mod 4 (6)

for all α ∈ R(�). Therefore, λτ (�) = −b1 + b3. Using the identification ofTheorem 4, we obtain the relation −b1 + b3 = µ̄(�). This completes the proof ofTheorem 5 in the case of n = 3.

If n � 4, the representation space R(�(a1, . . . , an)) is no longer nondegener-ate, and one needs perturbations in order to define and compute the Floer homol-ogy. We refer the reader to [26] for a complete account. ✷COROLLARY 6. Given a Seifert fibered homology sphere �(a1, . . . , an) let q =a1 · · · aj and p = aj+1 · · · an be products of the first j , respectively, last n − j ,Seifert invariants, where j is any integer such that 2 � j � n − 2. Then

I∗(�(a1, . . . , an)) = I∗(�(a1, . . . , aj , p)) ⊕ I∗(�(q, aj+1, . . . , an)).

Proof. The homology sphere �(a1, . . . , an) is the splice of �(a1, . . . , aj , p)

and �(q, aj+1, . . . , an) along singular fibers of degrees p and q, see for instance[8]. Both λ- and µ̄-invariants are additive with respect to splicing therefore so arethe Floer homology groups of �(a1, . . . , an). ✷

It is not known if Theorem 5 or splicing additivity hold for Floer homology ofgeneral graph homology spheres, or even just links of singularities.

Remark. Theorem 5 implies that the Floer homology of a Seifert homologysphere is completely determined by the topology of its Milnor fiber and its canon-ical resolution (which essentially answers the question posed by M. Atiyah, see[22], Question 3.5). It is not known if a similar statement holds for the Floer ho-mology of arbitrary links of singularities but resolving this issue in positive mightinclude proving the Neumann–Wahl conjecture as a first step.

3.3. RELATION TO HOMOLOGY COBORDISMS

An integral homology sphere � is said to be homology cobordant to zero if itbounds a smooth compact manifold X such that H∗(X,Z) = H∗(�,Z). Not everyintegral homology sphere is homology cobordant to zero – a classical obstructionis the Rohlin invariant which takes values in Z/2. In general, deciding whethera given homology sphere is homology cobordant to zero is an important but ex-tremely difficult problem in topology. The equivariant Casson invariant provides apartial solution in the case of links of singularities.


THEOREM 7. Suppose that an integral homology sphere � is a link of singular-ity, and let τ be the complex conjugation involution. If there exists an integer m

such that the connected sum of m copies of � is homology cobordant to zero thenλτ (�) � 0.

This theorem is proved in [25]. A stronger result holds for Seifert fibered homol-ogy spheres, see [24]. It uses Seiberg–Witten gauge theory on orbifolds developedin [12].

THEOREM 8. Let � = �(a1, . . . , an) be a Seifert fibered homology sphere andτ the complex conjugation involution. If there exists an integer m such that theconnected sum of m copies of � is homology cobordant to zero then λτ (�) = 0.

In particular, no multiple of a Seifert fibered homology sphere with nontrivialRohlin invariant can be homology cobordant to zero.

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