Complex surface singularities with integral homology sphere links

By Walter D. Neumann, J. Wahl. 2003

70 citations recorded by the source corpus.

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 [Geom. Topol. 9(2005) 699-755] 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 Sigma should be a 4-manifold canonically associated to Sigma. 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 [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann. 326(2003) 75--93].

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dbb37c4db3eab1f58da01be0958ab241e854b131

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In Geometry and Topology

1. A. N'emethi
2. J. Wahl
3. Walter D. Neumann
4. Tomohiro Okuma
5. A. Stipsicz
6. Zolt'an Szab'o
7. Stavros Garoufalidis
8. Don Zagier

In Mathematical Physics

1. A. N'emethi
2. J. Wahl
3. Walter D. Neumann
4. A. Stipsicz
5. Tomohiro Okuma
6. Zolt'an Szab'o
7. Stavros Garoufalidis
8. Don Zagier

Citations in the corpus

• Complete intersection singularities of splice type as universal abelian covers (2004, methodology): “s. (Sometimes —for instance the next section — it will be more convenient to move the last two columns of the above matrix to the first two columns.) 3 Splicediagram curves andassociated gradeds As in [11], one can define curves using a modified version of splice diagrams. Let (∆,w′) be a splice diagra”
• On the Casson Invariant Conjecture of Neumann–Wahl (2006, unknown): “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’).”
• Links and analytic invariants of superisolated singularities (2003, unknown): “C4 [ 33 ] , [2] T rue C5 [32,2] , [3] T rue C6 [32] , [3,2] T rue C7 [4,23] , [2] T rue C8 [4,22] , [22] T rue C9 [4] , [24] F alse (18) (3) N = 3 type of cusps conj”
• GRADED ROOTS AND SINGULARITIES (2007, result): “For related constructions and results, see also the articles of Neumann and Wahl [44, 45, 46].”