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A non-fixed point theorem for Hamiltonian lie group actions

Authors: Christopher Allday, Volker Hauschild and Volker Puppe
Journal: Trans. Amer. Math. Soc. 354 (2002), 2971-2982
MSC (2000): Primary 57S15; Secondary 53D99, 55N91, 57R17
Published electronically: March 5, 2002
MathSciNet review: 1895212
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Abstract: We prove that, under certain conditions, if a compact connected Lie group acts effectively on a closed manifold, then there is no fixed point. Because two of the main conditions are satisfied by any Hamiltonian action on a closed symplectic manifold, the theorem applies nicely to such actions. The method of proof, however, is cohomological; and so the result applies more generally.

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  • [Allday1] C. Allday, Examples of circle actions on symplectic spaces, Homotopy and Geometry, J. Oprea and A. Tralle, eds., Banach Center Publications 45, 87-90, Polish Academy of Sciences, Warsaw, 1998. MR 99k:57001
  • [Allday2] C. Allday, Notes on the Localization Theorem with applications to symplectic torus actions, Proceedings of the Winter School on Transformation Groups, Indian Statistical Institute, Calcutta 1998, to appear.
  • [Allday,Puppe] C. Allday and V. Puppe, Cohomological Methods in Transformation Groups, Cambridge Studies in Advanced Mathematics 32, Cambridge University Press, Cambridge, 1993. MR 94g:55009
  • [Atiyah,Bott] M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. MR 85e:58041
  • [Audin] M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics 93, Birkhäuser-Verlag, Basel, Boston, Berlin, 1991. MR 92m:57046
  • [Blanchard] A. Blanchard, Sur les variétés analytiques complexes, Annales Ecole Norm. Sup. 73 (1957), 157-202. MR 19:316e
  • [Borel et al.] A. Borel et al., Seminar on Transformation Groups, Ann. of Math. Studies 46, Princeton University Press, Princeton, 1960. MR 22:7129
  • [Bredon1] G. Bredon, Fixed point sets of actions on Poincaré duality spaces, Topology 12 (1973), 159-175. MR 48:9708
  • [Bredon2] G. Bredon, Sheaf Theory, Second Edition, Graduate Texts in Mathematics 170, Springer-Verlag, New York, 1997. MR 98g:55005
  • [Bredon3] G. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York (1972). MR 54:1265
  • [Chang] T. Chang, On the number of relations in the cohomologyof a fixed point set, Manuscript Math. 18 (1976), 237-247. MR 54:13954
  • [Chang,Skjelbred1] T. Chang and T. Skjelbred, Group actions on Poincaré duality spaces, Bull. Amer. Math. Soc. 78 (1972), 1024-1026. MR 46:6346
  • [Chang,Skjelbred2] T. Chang and T. Skjelbred, Lie group actions on a Cayley projective plane and a note on homogeneous spaces of prime Euler Characteristic, Amer. J. Math. 98 (1976), 655-678. MR 55:1371
  • [Frankel] T. Frankel, Fixed points and torsion on Kähler manifolds, Ann. of Math. 70 (1959), 1-8. MR 24:A1730
  • [Guillemin,Sternberg] V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer-Verlag, Berlin, Heidelberg, 1999. MR 2001i:53140
  • [Hauschild] V. Hauschild, Transformation groups on complex Grassmannians, to appear.
  • [Hsiang] W.-Y. Hsiang, Cohomology Theory of Topological Transformation Groups, Ergebnisse der Math. und ihrer Grenzgebiete 85, Springer-Verlag, New York, Heidelberg, Berlin, 1975. MR 54:11363
  • [Lupton,Oprea] G. Lupton and J. Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995), 261-288. MR 95f:57056
  • [Mann] L. Mann, Finite orbit structure on locally compact manifolds, Mich. Math. J. 9 (1962), 87-92. MR 24:A1966
  • [Quillen] D. Quillen, The spectrum of an equivariant cohomology ring: I, Ann. of Math. 94 (1971), 549-572. MR 45:7743
  • [Tolman,Weitsman] S. Tolman and J. Weitsman, The cohomologyrings of Abelian symplectic quotients,, July 1998. MR 98k:53038
  • [Tralle,Oprea] A. Tralle and J. Oprea, Symplectic Manifolds with no Kähler Structure, Lecture Notes in Math. 1661, Springer-Verlag, Berlin, Heidelberg, New York, 1997.

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Additional Information

Christopher Allday
Affiliation: Department of Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822-2273

Volker Hauschild
Affiliation: Department of Mathematics, University of Calabria, I-87036 Rende, Italy

Volker Puppe
Affiliation: Faculty of Mathematics, University of Konstanz, D-78457 Konstanz, Germany

Keywords: Compact connected Lie group actions, Hamiltonian actions, fixed points, cohomology theory
Received by editor(s): November 4, 2001
Published electronically: March 5, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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