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The Milnor Fiber Conjecture and iterated branched cyclic covers

Author: P. J. Lamberson
Journal: Trans. Amer. Math. Soc. 361 (2009), 4653-4681
MSC (2000): Primary 14B05, 32S55; Secondary 32S25, 32S50, 57M12, 57N10
Published electronically: April 15, 2009
MathSciNet review: 2506423
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Abstract: In this paper we prove the Milnor Fiber Conjecture of Neumann and Wahl for a class of isolated complete intersection singularities obtained by taking iterated branched cyclic covers of the singularity link. We also show that if the Milnor Fiber Conjecture holds for a given splice diagram, then it holds for any equivalent diagram satisfying the semigroup condition. We illustrate the application of these theorems in an example and discuss the relationship of these singularities with Neumann and Wahl's Splice Type Conjecture.

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  • 1. Olivier Collin and Nikolai Saveliev, Equivariant Casson invariants via gauge theory, J. Reine Agnew. Math. 541 (2001), 143-169. MR 1876288 (2002k:57077)
  • 2. David Eisenbud and Walter Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, 1985. MR 817982 (87g:57007)
  • 3. R. Fintushel and R. Stern, Instanton homology groups of Seifert fibered homology three spheres, Proc. London Math. Soc. 61 (1990), 109-137. MR 1051101 (91k:57029)
  • 4. Helmut Hamm, Lokale topologische Eigenschaften komplexer Ráume, Math. Ann. 191 (1971), 235-252. MR 0286143 (44:3357)
  • 5. -, Exotische Sphären als Umgebungsränder in speziellen komplexen Räuman, Math. Ann. 197 (1972), 44-56. MR 0314073 (47:2625)
  • 6. Louis Kauffman and Walter Neumann, Products of knots, branched fibrations, and sums of singularities, Topology 16 (1977), 369-393. MR 0488073 (58:7644)
  • 7. E. J. N. Looijenga, Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series, vol. 77, Cambridge University Press, Cambridge, 1984. MR 747303 (86a:32021)
  • 8. I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, Links and analytic invariants of superisolated singularities, J. Algebraic Geom. 14 (2005), 543-565. MR 2129010 (2005m:32057)
  • 9. John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, vol. 61, Princeton University Press, Princeton, 1968. MR 0239612 (39:969)
  • 10. András Némethi, Invariants of normal surface singularities, Real and complex singularities, Contemp. Math., vol. 354, Amer. Math. Soc., Providence, RI, 2004, pp. 161-208. MR 2087811 (2005g:32040)
  • 11. András Némethi and Liviu I. Nicolaescu, Seiberg-Witten invariants and surface singularities III: Splicings and cyclic covers.
  • 12. -, Seiberg-Witten invariants and surface singularities, Geom. Topol. 6 (2002), 269-328. MR 1914570 (2003i:14048)
  • 13. -, Seiberg-Witten invariants and surface singularities II: Singularities with good $ {\mathbb{C}^*}$-action, J. London Math. Soc. 69 (2004), 593-607. MR 2050035 (2005g:14070)
  • 14. Walter D. Neumann, Abelian covers of quasihomogeneous surface singularities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 233-243. MR 713252 (85g:32018)
  • 15. Walter D. Neumann and Frank Raymond, Seifert manifolds, plumbing, $ \mu$-invariant and orientation reversing maps, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977), Lecture Notes in Math., vol. 664, Springer, Berlin, 1978, pp. 163-196. MR 518415 (80e:57008)
  • 16. Walter D. Neumann and Jonathan Wahl, Casson invariant of links of singularities, Comment. Math. Helv. 65 (1990), no. 1, 58-78. MR 1036128 (91c:57022)
  • 17. -, Universal abelian covers of quotient-cusps, Math. Ann. 326 (2003), no. 1, 75-93. MR 1981612 (2004d:32039)
  • 18. -, Complete intersection singularities of splice type as universal abelian covers, Geom. Topol. 9 (2005), 699-755 (electronic). MR 2140991
  • 19. -, Complex surface singularities with integral homology sphere links, Geom. Topol. 9 (2005), 757-811 (electronic). MR 2140992 (2006b:32042)
  • 20. Tomohiro Okuma, Universal abelian covers of certain surface singularities.

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Additional Information

P. J. Lamberson
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Address at time of publication: Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109

Received by editor(s): June 25, 2007
Published electronically: April 15, 2009
Additional Notes: This work was carried out while the author was a Ph.D. student at Columbia University, supported by a graduate fellowship. Part of this research was completed while the author was supported by the Carl B. Boyer Memorial Fellowship. The author wishes to thank Walter Neumann for invaluable guidance in conducting this research.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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