Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Smoothability of $ \mathbb{Z}\times\mathbb{Z}$-actions on 4-manifolds


Author: Nobuhiro Nakamura
Journal: Proc. Amer. Math. Soc. 138 (2010), 2973-2978
MSC (2010): Primary 57S05; Secondary 57M60, 57R57
DOI: https://doi.org/10.1090/S0002-9939-10-10413-4
Published electronically: March 23, 2010
MathSciNet review: 2644908
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a nonsmoothable $ \mathbb{Z}\times\mathbb{Z}$-action on the connected sum of an Enriques surface and $ S^2\times S^2$, such that each of the generators is smoothable. We also construct a nonsmoothable self-homeomorphism on an Enriques surface.


References [Enhancements On Off] (What's this?)

  • 1. A. Edmonds, Aspects of group actions on four-manifolds, Topology Appl. 31 (1989), no. 2, 109-124. MR 994404 (90h:57050)
  • 2. M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Mathematical Series, 39. Princeton University Press, Princeton, NJ, 1990. MR 1201584 (94b:57021)
  • 3. I. Hambleton and M. Kreck, Smooth structures on algebraic surfaces with cyclic fundamental group, Invent. Math. 91 (1988), no. 1, 53-59. MR 918236 (89a:57043)
  • 4. R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations. Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1977. MR 0645390 (58:31082)
  • 5. N. Nakamura, The Seiberg-Witten equations for families and diffeomorphisms of $ 4$-manifolds, Asian J. Math. 7 (2003), no. 1, 133-138. MR 2015245 (2004h:57037)
  • 6. -, Correction, Asian J. Math. 9 (2005), no. 2, 185. MR 2176601 (2006e:57040)
  • 7. D. Ruberman, An obstruction to smooth isotopy in dimension $ 4$, Math. Res. Lett. 5 (1998), no. 6, 743-758. MR 1671187 (2000c:57061)
  • 8. D. Ruberman, Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants, Geom. Topol. 5 (2001), 895-924. MR 1874146 (2002k:57076)
  • 9. M. Szymik, Characteristic cohomotopy classes for families of $ 4$-manifolds, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57S05, 57M60, 57R57

Retrieve articles in all journals with MSC (2010): 57S05, 57M60, 57R57


Additional Information

Nobuhiro Nakamura
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan
Email: nobuhiro@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-10-10413-4
Keywords: Group action, smoothability, Enriques surface
Received by editor(s): February 23, 2009
Received by editor(s) in revised form: November 22, 2009
Published electronically: March 23, 2010
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society