Equivariant Casson invariants via gauge theory

Collin, Olivier; Saveliev, Nikolai

doi:10.1515/crll.2001.090

Cited by 24 publications

(70 citation statements)

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“…Though the Casson Invariant Conjecture for this case follows, it had already been proven in [26] (by a much less conceptual proof), and more recently by Collin and Saveliev [5] using equivariant Casson invariants and by Némethi and Nicolaescu [23] in a more general context. It is also a special case of Theorem 2.…”

Section: Theoremmentioning

confidence: 84%

“…The results of this section give a proof of the Casson Invariant Conjecture (CIC) for these examples, also proven in [26,5,23]. Saveliev and Collin [5], using equivariant Casson invariant, give an iterative generalization of these examples but their approach implies more: Let ∆ be any splice diagram satisfying the semigroup condition and w a leaf of ∆.…”

Section: Remark 83mentioning

confidence: 95%

“…Saveliev and Collin [5], using equivariant Casson invariant, give an iterative generalization of these examples but their approach implies more: Let ∆ be any splice diagram satisfying the semigroup condition and w a leaf of ∆. We allow, as in this section, the weight on the edge to w to be 1.…”

Section: Remark 83mentioning

confidence: 99%

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Complex surface singularities with integral homology sphere links

Neumann

Wahl

2005

Geom. Topol.

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While the topological types of normal surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [30] that many of them can be realized as complete intersection singularities of "splice type," generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link Σ should be a 4-manifold canonically associated to Σ. We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We also point out recent counterexamples to some overly optimistic earlier conjectures in [28] and [29]. In the parallel paper [30] we give analytic descriptions in terms of splice diagrams for a wide range of topologies of singularities, when the link of the singularity is a Q-homology sphere. The splice diagrams considered there generalize the original splice diagrams of [7,31] in that the numerical weights around a node need not be pairwise coprime. In this paper we restrict to Zhomology sphere links. Our splice diagrams will thus always have pairwise coprime weights around each node, and, by [7] There is a natural notion of "higher weight terms" for a splice type equation, and, by definition, the result of adding higher weight terms is still of splice type 1 (the effect on the singularity is always an equisingular deformation). Thus, for example, the splice type singularities corresponding to one-node splice diagrams are precisely the Brieskorn complete intersection singularities with homology sphere link and their higher weight deformations. AMS Classification numbersIn an earlier paper [28], we made the over-optimistic Splice Type Conjecture Any Gorenstein surface singularity with integral homology sphere link is a complete intersection of splice type.Implicit in this conjecture was a new necessary condition (the "semigroup condition") on a splice diagram (and hence on a resolution diagram) in order that 1 This differs from [28,29], where higher order terms were not allowed. We now call this "strict splice type."Geometry & Topology, Volume 9 (2005) Complex surface singularities with integral homology sphere links 759 it come from a Gorenstein singularity. After all, a similar semigroup condition on the value semigroup of a curve singularity is well known to characterize the Gorenstein ones. Further, ...

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Section: Theoremmentioning

confidence: 84%

Section: Remark 83mentioning

confidence: 95%

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Complex surface singularities with integral homology sphere links

Neumann

Wahl

2005

Geom. Topol.

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“…The author was informed by Saveliev that the parallels on the critical space level can be explained by results of Collin and Saveliev [2] and Saveliev [16; 17]. For knots the representation spaces as considered here correspond to Z=2 equivariant representation spaces of the manifold obtained as double branched cover of the knot complement, branched along the knot [2,Proposition 3.3].…”

Section: Parallels With Representation Spaces Of Brieskorn Homology Smentioning

confidence: 99%

Representation spaces of pretzel knots

Zentner

2011

Algebr. Geom. Topol.

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We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not binary dihedral.Comment: 30 pages, 3 figures. This is the published version. It is more general than the previous version regarding the values $p,q,r$. Furthermore, rewritten according after referee's remark

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“…Some iterative generalizations, related with cyclic coverings and using techniques of equivariant Casson invariant and gauge theory, were covered by Collin and Saveliev (cf. [3,4,5]). …”

Section: An Isolated Complete Intersection Surface Singularity Whose mentioning

confidence: 99%

On the Casson Invariant Conjecture of Neumann–Wahl

Némethi

Okuma

2008

J. Algebraic Geom.

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Abstract. In the article we prove the Casson Invariant Conjecture of NeumannWahl for splice type surface singularities. Namely, for such an isolated complete intersection, whose link is an integral homology sphere, we show that the Casson invariant of the link is one-eighth the signature of the Milnor fiber. IntroductionAlmost twenty years ago, Neumann and Wahl formulated the following conjecture: an isolated complete intersection surface singularity whose link Σ is an integral homology 3-sphere Then the Casson invariant λ(Σ) of the link is one-eighth the signature of the Milnor fiber of (X, o).The conjecture can be reformulated in terms of the geometric genus pg of (X, o) as well (see below in 4.1).The conjecture is true for Brieskorn hypersurface singularities by a result of Fintushel and Stern [8]. This and additivity properties (with respect to splice decomposition) lead to the verification of the conjecture for Brieskorn complete intersections, done independently by Neumann-Wahl [15] and Fukuhara-Matsumoto-Sakamoto [9]. For suspension hypersurface singularities it was verified in [15]. Some iterative generalizations, related with cyclic coverings and using techniques of equivariant Casson invariant and gauge theory, were covered by Collin and Saveliev (cf. [3,4,5]).Recently, Neumann and Wahl have introduced an important family of complete intersection surface singularities, the splice type singularities. In [18] they treated the case when the link is an integral homology sphere (the reader may consult in [16,17,19] the case of rational homology sphere links too). In [18], they have also verified the above conjecture (by a direct computation of the geometric genus) for special splice type singularities (when the nodes of the splice diagram 'are in a line').The goal of the present article is to verify the conjecture for an arbitrary splice type singularity:Theorem. The Casson Invariant Conjecture is true for any splice type singularity with integral homology sphere link.

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Equivariant Casson invariants via gauge theory

Cited by 24 publications

References 18 publications

Complex surface singularities with integral homology sphere links

Complex surface singularities with integral homology sphere links

Representation spaces of pretzel knots

On the Casson Invariant Conjecture of Neumann–Wahl

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