Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

On the $ \overline{\mu}$-invariant of rational surface singularities

Author(s): András I. Stipsicz
Journal: Proc. Amer. Math. Soc. 136 (2008), 3815-3823.
MSC (2000): Primary 14J17, 57M27
Posted: May 28, 2008
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

1.
A. Casson and J. Harer, Some homology lens spaces which bound rational homology balls, Pacific J. Math. 96 (1981) 23-36. MR 634760 (83h:57013)

2.
R. Fintushel and R. Stern, Rational blowdowns of smooth $ 4$-manifolds, J. Diff. Geom. 46 (1997) 181-235. MR 1484044 (98j:57047)

3.
R. Fintushel and R. Stern, A $ \mu$-invariant one homology $ 3$-sphere that bounds an orientable rational ball, Contemporary Math. 35, AMS, 1984, 265-268. MR 780582 (86f:57013)

4.
D. Galewski and R. Stern, Classification of simplicial triangulations of topological manifolds, Ann. Math. (2) 111 (1980) 1-34. MR 558395 (81f:57012)

5.
R. Gompf and A. Stipsicz, $ 4$-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, AMS, 1999. MR 1707327 (2000h:57038)

6.
H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962) 331-368. MR 0137127 (25:583)

7.
J. Greene and S. Jabuka, The slice-ribbon conjecture for $ 3$-stranded pretzel knots, arXiv:0706.3398.

8.
S. Jabuka and S. Naik, Order in the concordance group and Heegaard Floer homology, Geom. Topol. 11 (2007) 979-994. MR 2326940

9.
P. Lisca and A. Stipsicz, Ozsváth-Szabó invariants and tight contact $ 3$-manifolds, I, Geom. Topol. 8 (2004) 925-945. MR 2087073 (2005e:57069)

10.
P. Lisca and A. Stipsicz, Ozsváth-Szabó invariants and tight contact $ 3$-manifolds, II, J. Differential Geom. 75 (2007) 109-141. MR 2282726

11.
P. Lisca and A. Stipsicz, Ozsváth-Szabó invariants and tight contact $ 3$-manifolds, III, J. Symplectic Geometry, to appear, arXiv:math.SG/0505493.

12.
P. Lisca and A. Stipsicz, On the existence of tight contact structures on Seifert fibered $ 3$-manifolds, arXiv:0709.0737

13.
A. Némethi, On the Ozsváth-Szabó invariant of negative definite plumbed $ 3$-manifolds, Geom. Topol. 9 (2005) 991-1042. MR 2140997 (2006c:57011)

14.
W. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen, 1979, 125-144. Lect. Notes in Math. 788, Springer, Berlin, 1980. MR 585657 (82j:57033)

15.
W. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981) 299-344. MR 632532 (84a:32015)

16.
P. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179-261. MR 1957829 (2003m:57066)

17.
P. Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027-1158. MR 2113019 (2006b:57016)

18.
P. Ozsváth and Z. Szabó, Holomorphic disks and three-manifold invariants: Properties and applications, Ann. of Math. 159 (2004) 1159-1245. MR 2113020 (2006b:57017)

19.
P. Ozsváth and Z. Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003) 185-224. MR 1988284 (2004h:57039)

20.
J. Park, Seiberg-Witten invariants of generalized rational blow-downs, Bull. Austral. Math. Soc. 56 (1997) 363-384. MR 1490654 (99b:57067)

21.
N. Saveliev, Fukumoto-Furuta invariants of plumbed homology $ 3$-spheres, Pacific J. Math. 205 (2002) 465-490. MR 1922741 (2003k:57038)

22.
A. Stipsicz, Z. Szabó and J. Wahl, Rational blow-downs and smoothings of surface singularities, J. Topology 1 (2008) 477-517.

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14J17, 57M27

Retrieve articles in all Journals with MSC (2000): 14J17, 57M27

Additional Information:

András I. Stipsicz
Affiliation: Rényi Institute of Mathematics, H-1053 Budapest, Reáltanoda utca 13--15, Hungary - and - Department of Mathematics, Columbia University, New York, New York 10027
Email: stipsicz@math-inst.hu, stipsicz@math.columbia.edu

DOI: 10.1090/S0002-9939-08-09439-2
PII: S 0002-9939(08)09439-2
Received by editor(s): September 28, 2007
Posted: May 28, 2008
Communicated by: Daniel Ruberman
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google