On the \overline{𝜇}–invariant of rational surface singularities

Stipsicz, András

doi:10.1090/S0002-9939-08-09439-2

On the $\overline {\mu }$–invariant of rational surface singularities
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Proc. Amer. Math. Soc. 136 (2008), 3815-3823 Request permission

Abstract:

We show that for rational surface singularities with odd determinant the $\overline {\mu }$–invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4–ball. We identify the $\overline {\mu }$–invariant with the corresponding correction term in Heegaard Floer theory.

References

Andrew J. Casson and John L. Harer, Some homology lens spaces which bound rational homology balls, Pacific J. Math. 96 (1981), no. 1, 23–36. MR 634760, DOI 10.2140/pjm.1981.96.23
Ronald Fintushel and Ronald J. Stern, Rational blowdowns of smooth $4$-manifolds, J. Differential Geom. 46 (1997), no. 2, 181–235. MR 1484044
Ronald Fintushel and Ronald J. Stern, A $\mu$-invariant one homology $3$-sphere that bounds an orientable rational ball, Four-manifold theory (Durham, N.H., 1982) Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 265–268. MR 780582, DOI 10.1090/conm/035/780582
David E. Galewski and Ronald J. Stern, Classification of simplicial triangulations of topological manifolds, Ann. of Math. (2) 111 (1980), no. 1, 1–34. MR 558395, DOI 10.2307/1971215
Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
Hans Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368 (German). MR 137127, DOI 10.1007/BF01441136
J. Greene and S. Jabuka, The slice–ribbon conjecture for $3$–stranded pretzel knots, arXiv:0706.3398.
Stanislav Jabuka and Swatee Naik, Order in the concordance group and Heegaard Floer homology, Geom. Topol. 11 (2007), 979–994. MR 2326940, DOI 10.2140/gt.2007.11.979
Paolo Lisca and András I. Stipsicz, Ozsváth-Szabó invariants and tight contact three-manifolds. I, Geom. Topol. 8 (2004), 925–945. MR 2087073, DOI 10.2140/gt.2004.8.925
Paolo Lisca and András I. Stipsicz, Ozsváth-Szabó invariants and tight contact three-manifolds. II, J. Differential Geom. 75 (2007), no. 1, 109–141. MR 2282726
P. Lisca and A. Stipsicz, Ozsváth–Szabó invariants and tight contact $3$–manifolds, III, J. Symplectic Geometry, to appear, arXiv:math.SG/0505493.
P. Lisca and A. Stipsicz, On the existence of tight contact structures on Seifert fibered $3$–manifolds, arXiv:0709.0737
András Némethi, On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds, Geom. Topol. 9 (2005), 991–1042. MR 2140997, DOI 10.2140/gt.2005.9.991
Walter D. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 125–144. MR 585657
Walter D. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), no. 2, 299–344. MR 632532, DOI 10.1090/S0002-9947-1981-0632532-8
Peter Ozsváth and Zoltán Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179–261. MR 1957829, DOI 10.1016/S0001-8708(02)00030-0
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. MR 2113020, DOI 10.4007/annals.2004.159.1159
Peter Ozsváth and Zoltán Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003), 185–224. MR 1988284, DOI 10.2140/gt.2003.7.185
Jongil Park, Seiberg-Witten invariants of generalised rational blow-downs, Bull. Austral. Math. Soc. 56 (1997), no. 3, 363–384. MR 1490654, DOI 10.1017/S0004972700031154
Nikolai Saveliev, Fukumoto-Furuta invariants of plumbed homology 3-spheres, Pacific J. Math. 205 (2002), no. 2, 465–490. MR 1922741, DOI 10.2140/pjm.2002.205.465
A. Stipsicz, Z. Szabó and J. Wahl, Rational blow–downs and smoothings of surface singularities, J. Topology 1 (2008) 477–517.