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Transactions of the American Mathematical Society

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The cohomology representation of an action of $ C\sb p$ on a surface


Author: Peter Symonds
Journal: Trans. Amer. Math. Soc. 306 (1988), 389-400
MSC: Primary 57S17; Secondary 20C10, 57M12
DOI: https://doi.org/10.1090/S0002-9947-1988-0927696-9
MathSciNet review: 927696
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Abstract: When a finite group $ G$ acts on a surface $ S$, then $ {H^1}(S;\,{\mathbf{Z}})$ posseses naturally the structure of a $ {\mathbf{Z}}G$-module with invariant symplectic inner product. In the case of a cyclic group of odd prime order we describe explicitly this symplectic inner product space in terms of the fixed-point data of the action.


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  • [1] J. F. Adams, Lectures on Lie groups, Benjamin, New York, 1969. MR 0252560 (40:5780)
  • [2] J. P. Alexander, P. E. Conner and G. C. Hamrick, Odd order group actions and Witt classification of inner products, Lecture Notes in Math. vol. 625, Springer, Berlin, Heidelberg and New York, 1963.
  • [3] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546-604. MR 0236952 (38:5245)
  • [4] P. E. Conner, Notes on the Witt classification of Hermitian innerproduct spaces over a ring of algebraic integers, Univ. of Texas Press, Austin, 1979. MR 551205 (81i:10026)
  • [5] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Wiley, New York, 1962. MR 0144979 (26:2519)
  • [6] C. J. Earle, On the moduli of closed Riemann surfaces with symmetries, Ann. of Math. Studies, no. 66, Princeton Univ. Press, Princeton, N.J., 1971. MR 0296282 (45:5343)
  • [7] A. L. Edmonds and J. H. Ewing, Surface symmetry and homology, Math. Proc. Cambridge Philos. Soc. 99 (1986), 73-77. MR 809500 (87a:57045)
  • [8] J. H. Ewing, The image of the Atiyah-Bott map, Math. Z. 165 (1979), 53-71. MR 521520 (81g:57026)
  • [9] H. M. Farkas and I. Kra, Riemann surfaces, Springer, New York, Heidelberg and Berlin, 1980. MR 583745 (82c:30067)
  • [10] H. H. Glover and G. Mislin, Torsion in the mapping class group and its cohomology, J. Pure Appl. Algebra 44 (1987), 177-189. MR 885103 (88d:57001)
  • [11] S. Lang, Algebraic number theory, Addition-Wesley, Reading, Mass., 1970. MR 0282947 (44:181)
  • [12] J. Nielsen, Die Sturctur periodischer Transformationer von Flächen, Danske Vid. Selsk. Mat.-Fys. Medd. 15 (1937), 1-77.
  • [13] -, Abbildungsklassen endlicher Ordnung, Acta Math. 75 (1943), 23-115. MR 0013306 (7:137a)
  • [14] O. T. O'Meara, Introduction to quadratic forms, Grundlehren Math. Wiss., vol. 117, Springer, Berlin, Heidelberg and New York, 1963. MR 0152507 (27:2485)
  • [15] R. G. Swan, Induced representations and projective modules, Ann. of Math. (2) 71 (1960), 552-578. MR 0138688 (25:2131)
  • [16] H. Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Math., vol. 875, Springer, Berlin, Heidelberg and New York, 1981. MR 643627 (86g:57001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0927696-9
Article copyright: © Copyright 1988 American Mathematical Society

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