Symplectic group actions on homotopy elliptic surfaces

Abstract

In this paper, the authors study the homologically trivial symplectic group actions on homotopy elliptic surfaces E(n) and get some rigidity results.

Keywords

Symplectic action / Homologically trivial / g-Signature theorem / Fixed-point data

References

[1] Atiyah M F, Bott R. A Lefschetz fixed point formula for elliptic complexes: II. Application. Ann. of Math., 1968, 88: 451-491

[2] Barth W, Peter C, Van de Ven A. Compact Complex Surfaces, 1984, Berlin, Heidelberg, New York, Tokyo: Springer-Verlag

[3] Chen W. Group actions on 4-manifolds: Some recent results and open questions. Proceedings of 13^th Gökova Geometry-Topology Conference, 2009 1-21

[4] Chen W, Kwasik S. Symplectic symmetries of 4-manifolds. Topology, 2007, 46(2): 103-128

[5] Edmonds A L. Construction of group actions on four-manifold. Trans. Amer. Math. Soc., 1987, 299: 155-170

[6] Edmonds, A. L., A survey of group actions on 4-manifolds, arXiv: 0907.0454.

[7] Edmonds A L, Ewing J H. Realizing forms and fixed point data in dimension four. Amer. J. Math., 1992, 114: 1103-1126

[8] Fintushel R, Stern R. Knots, links and 4-manifolds. Invent. Math., 1998, 134: 363-400

[9] Gompf R E, Stipsicz A I. 4-Manifolds and Kirby Calculus, 1999, Providence, RI: Amer. Math. Soc.

[10] Gordon C McA. On the g-signature theorem in dimension four. A la Recherche de la Topologie Perdue, Progress in Math., 1986, 62: 159-180

[11] Hirzebruch F, Zagier D. The Atiyah-Singer Theorem and Elementary Number Theory, 1974

[12] Kwasik S, Schultz R. Homological properties of periodic homeomorphisms of 4-manifolds. Duke Math. J., 1989, 58: 241-250

[13] Liu X, Nakamura N. Pseudofree $\tfrac{Z}{3}$-actions on K3 surfaces. Proc. Amer. Math. Soc., 2007, 135(3): 903-910

[14] Liu X, Nakamura N. Nonsmoothable group actions on elliptic surfaces. Topol. Appl., 2008, 155: 946-964

[15] Matumoto T. Homologically trivial smooth involutions on K3 surfaces, Aspects of Low-dimensional Manifolds. Advanced Studies in Pure Mathematics, 1992, 20: 365-376

[16] McCooey M. Symmetry groups of four-manifolds. Topology, 2002, 41(4): 835-851

[17] Orlik P. Seifert manifolds, 1972, Berlin, Heidelberg, New York: Springer-Verlag

[18] Peters C A M. On automorphisms of compact Kähler surfaces, 1979, Angers: Algebraic Geometry 249-267

[19] Ruberman D. Involutions on spin 4-manifolds. Proc. Aemr. Math. Soc., 1995, 123(2): 593-596

[20] Taubes C. The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett., 1994, 1: 809-822

[21] Taubes C. SW ⇒ Gr: From the Seiberg-Witten equations to pseudohollomorphic curves. J. Amer. Math. Soc., 1996, 9: 845-918