## Cyclic group actions and embedded spheres in 4-manifolds

HTML articles powered by AMS MathViewer

- by M. J. D. Hamilton
- Proc. Amer. Math. Soc.
**144**(2016), 411-422 - DOI: https://doi.org/10.1090/proc/12707
- Published electronically: July 24, 2015
- PDF | Request permission

## Abstract:

In this note we derive an upper bound on the number of 2-spheres in the fixed point set of a smooth and homologically trivial cyclic group action of prime order on a simply-connected 4-manifold. This improves the a priori bound which is given by one half of the Euler characteristic of the 4-manifold. The result also shows that in some cases the 4-manifold does not admit such actions of a certain order at all or that any such action has to be pseudofree.## References

- Anar Akhmedov,
*Small exotic 4-manifolds*, Algebr. Geom. Topol.**8**(2008), no. 3, 1781–1794. MR**2448872**, DOI 10.2140/agt.2008.8.1781 - Anar Akhmedov and B. Doug Park,
*Exotic smooth structures on small 4-manifolds with odd signatures*, Invent. Math.**181**(2010), no. 3, 577–603. MR**2660453**, DOI 10.1007/s00222-010-0254-y - M. F. Atiyah and R. Bott,
*A Lefschetz fixed point formula for elliptic complexes. II. Applications*, Ann. of Math. (2)**88**(1968), 451–491. MR**232406**, DOI 10.2307/1970721 - M. F. Atiyah and I. M. Singer,
*The index of elliptic operators. III*, Ann. of Math. (2)**87**(1968), 546–604. MR**236952**, DOI 10.2307/1970717 - Glen E. Bredon,
*Introduction to compact transformation groups*, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR**0413144** - Dan Burns Jr. and Michael Rapoport,
*On the Torelli problem for kählerian $K-3$ surfaces*, Ann. Sci. École Norm. Sup. (4)**8**(1975), no. 2, 235–273. MR**447635** - Weimin Chen and Slawomir Kwasik,
*Symplectic symmetries of 4-manifolds*, Topology**46**(2007), no. 2, 103–128. MR**2313067**, DOI 10.1016/j.top.2006.12.003 - S. K. Donaldson,
*An application of gauge theory to four-dimensional topology*, J. Differential Geom.**18**(1983), no. 2, 279–315. MR**710056** - Allan L. Edmonds,
*Construction of group actions on four-manifolds*, Trans. Amer. Math. Soc.**299**(1987), no. 1, 155–170. MR**869405**, DOI 10.1090/S0002-9947-1987-0869405-7 - Allan L. Edmonds,
*Involutions on odd four-manifolds*, Topology Appl.**30**(1988), no. 1, 43–49. MR**964061**, DOI 10.1016/0166-8641(88)90079-X - Allan L. Edmonds,
*Aspects of group actions on four-manifolds*, Topology Appl.**31**(1989), no. 2, 109–124. MR**994404**, DOI 10.1016/0166-8641(89)90075-8 - 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 - F. Hirzebruch,
*The signature theorem: reminiscences and recreation*, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 3–31. MR**0368023** - Michael Klemm,
*Finite group actions on smooth 4-manifolds with indefinite intersection form*, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–McMaster University (Canada). MR**2695049** - D. Kotschick,
*Orientations and geometrisations of compact complex surfaces*, Bull. London Math. Soc.**29**(1997), no. 2, 145–149. MR**1425990**, DOI 10.1112/S0024609396002287 - P. B. Kronheimer and T. S. Mrowka,
*The genus of embedded surfaces in the projective plane*, Math. Res. Lett.**1**(1994), no. 6, 797–808. MR**1306022**, DOI 10.4310/MRL.1994.v1.n6.a14 - Sławomir Kwasik and Reinhard Schultz,
*Homological properties of periodic homeomorphisms of $4$-manifolds*, Duke Math. J.**58**(1989), no. 1, 241–250. MR**1016421**, DOI 10.1215/S0012-7094-89-05812-2 - Hongxia Li,
*Cyclic group actions on elliptic surfaces $E(2n)$*, J. Math. Comput. Sci.**2**(2012), no. 6, 1759–1765. MR**3004191** - T. J. Li and A. Liu,
*Symplectic structure on ruled surfaces and a generalized adjunction formula*, Math. Res. Lett.**2**(1995), no. 4, 453–471. MR**1355707**, DOI 10.4310/MRL.1995.v2.n4.a6 - Ai-Ko Liu,
*Some new applications of general wall crossing formula, Gompf’s conjecture and its applications*, Math. Res. Lett.**3**(1996), no. 5, 569–585. MR**1418572**, DOI 10.4310/MRL.1996.v3.n5.a1 - Takao Matumoto,
*Homologically trivial smooth involutions on $K3$ surfaces*, Aspects of low-dimensional manifolds, Adv. Stud. Pure Math., vol. 20, Kinokuniya, Tokyo, 1992, pp. 365–376. MR**1208316**, DOI 10.2969/aspm/02010365 - M. McCooey,
*Symmetry groups of non-simply connected four-manifolds*, preprint: arXiv:0707.3835v2. - C. A. M. Peters,
*On automorphisms of compact Kähler surfaces*, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 249–267. MR**605346** - Ulf Persson, Chris Peters, and Gang Xiao,
*Geography of spin surfaces*, Topology**35**(1996), no. 4, 845–862. MR**1404912**, DOI 10.1016/0040-9383(95)00046-1 - Daniel Ruberman,
*Involutions on spin $4$-manifolds*, Proc. Amer. Math. Soc.**123**(1995), no. 2, 593–596. MR**1231042**, DOI 10.1090/S0002-9939-1995-1231042-2 - P. A. Smith,
*Transformations of finite period. IV. Dimensional parity*, Ann. of Math. (2)**46**(1945), 357–364. MR**13304**, DOI 10.2307/1969155 - Zoltán Szabó,
*Exotic $4$-manifolds with $b^+_2=1$*, Math. Res. Lett.**3**(1996), no. 6, 731–741. MR**1426531**, DOI 10.4310/MRL.1996.v3.n6.a2 - C. T. C. Wall,
*Surgery on compact manifolds*, 2nd ed., Mathematical Surveys and Monographs, vol. 69, American Mathematical Society, Providence, RI, 1999. Edited and with a foreword by A. A. Ranicki. MR**1687388**, DOI 10.1090/surv/069 - Edward Witten,
*Monopoles and four-manifolds*, Math. Res. Lett.**1**(1994), no. 6, 769–796. MR**1306021**, DOI 10.4310/MRL.1994.v1.n6.a13

## Bibliographic Information

**M. J. D. Hamilton**- Affiliation: Institute for Geometry and Topology, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
- Email: mark.hamilton@math.lmu.de
- Received by editor(s): July 29, 2014
- Received by editor(s) in revised form: December 4, 2014
- Published electronically: July 24, 2015
- Communicated by: Martin Scharlemann
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 411-422 - MSC (2010): Primary 57M60, 57S17, 57N13; Secondary 57R57
- DOI: https://doi.org/10.1090/proc/12707
- MathSciNet review: 3415607