A Geometric Proof of the Fintushel–Stern Formula

Collin, Olivier; Saveliev, Nikolai

doi:10.1006/aima.1999.1842

Cited by 18 publications

(17 citation statements)

References 14 publications

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“…consistent with the orientation to identify Hom(7rdFo), SU(2» with SU(2)2 9 . Requiring that the map (3.6) be orientation-preserving orients the manifold R(F). Choose bases in 58 3 easson Invariant HI (MI, IR) and H I (M2 , IR) consistent with the above orientations and use them to orient R*(MI ) and R*(M2 ), respectively.…”

Section: Orientationsmentioning

confidence: 99%

Invariants for Homology 3-Spheres

Saveliev

2002

Encyclopaedia of Mathematical Sciences

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122

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Section: Orientationsmentioning

confidence: 99%

Invariants for Homology 3-Spheres

Saveliev

2002

Encyclopaedia of Mathematical Sciences

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122

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“…For this case, and under the assumption that v is a node, a pg-formula is established in [20]. 4. The proof of the main theorem 4.1.…”

Section: Note That If Two Formal Power Series F1(t) and F2(t) Have Pementioning

confidence: 95%

“…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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“…where F (p, q, r) is the Milnor fiber (see [68] for a geometric proof). This led to the Casson invariant conjecture stated in [205]:…”

Section: Low Dimensional Manifolds Ifmentioning

confidence: 99%

On Milnor’s fibration theorem and its offspring after 50 years

Seade

2018

Bull. Amer. Math. Soc.

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To Jack, whose profoundness and clarity of vision seep into our appreciation of the beauty and depth of mathematics.Abstract. Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is two-fold. On the one hand, it should serve as an introduction to the topic for non-experts, and on the other hand, it gives a wide perspective of some of the work on the subject that has been and is being done. It includes a vast literature for further reading. IntroductionMilnor's fibration theorem in [189] is a milestone in singularity theory that has opened the way to a myriad of insights and new understandings. This is a beautiful piece of mathematics, where many different branches, aspects and ideas, come together. The theorem concerns the geometry and topology of analytic maps near their critical points.Consider the simplest case, a holomorphic map (C n+1 , 0) f → (C, 0) taking the origin into the origin, with an isolated critical point at 0. As an example one can have in mind the Pham-Brieskorn polynomials:(0.1) z → z a 0 0 + · · · + z an n , a i ≥ 2 for all i = 0, 1, · · · , n .Since f is analytic, there exists r > 0 sufficiently small so that 0 ∈ C is the only critical value of the restriction f | Br , where B r is the open ball of radius r and center at 0. Set V := f −1 (0) and V * := V \ {0} .

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A Geometric Proof of the Fintushel–Stern Formula

Cited by 18 publications

References 14 publications

Invariants for Homology 3-Spheres

Invariants for Homology 3-Spheres

On the Casson Invariant Conjecture of Neumann–Wahl

On Milnor’s fibration theorem and its offspring after 50 years

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