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
DOI:
https://doi.org/10.1090/S0002-9947-02-02968-9
Published electronically:
March 5, 2002
MathSciNet review:
1895212
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- John Oprea and Aleksy Tralle (eds.), Homotopy and geometry, Banach Center Publications, vol. 45, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 1998. Papers from the workshop held in Warsaw, June 9–13, 1997. MR 1683657
- 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.
- C. Allday and V. Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics, vol. 32, Cambridge University Press, Cambridge, 1993. MR 1236839
- M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. MR 721448, DOI https://doi.org/10.1016/0040-9383%2884%2990021-1
- Michèle Audin, The topology of torus actions on symplectic manifolds, Progress in Mathematics, vol. 93, Birkhäuser Verlag, Basel, 1991. Translated from the French by the author. MR 1106194
- A. Blanchard, Sur les variétés analytiques complexes, Annales Ecole Norm. Sup. 73 (1957), 157–202.
- Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
- Glen E. Bredon, Fixed point sets of actions on Poincaré duality spaces, Topology 12 (1973), 159–175. MR 331375, DOI https://doi.org/10.1016/0040-9383%2873%2990004-9
- Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706
- Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
- Theodore Chang, On the number of relations in the cohomology of a fixed point set, Manuscripta Math. 18 (1976), no. 3, 237–247. MR 426005, DOI https://doi.org/10.1007/BF01245918
- Theodore Chang and Tor Skjelbred, Group actions on Poincaré duality spaces, Bull. Amer. Math. Soc. 78 (1972), 1024–1026. MR 307226, DOI https://doi.org/10.1090/S0002-9904-1972-13092-1
- Theodore Chang and Tor Skjelbred, Lie group actions on a Cayley projective plane and a note on homogeneous spaces of prime Euler characteristic, Amer. J. Math. 98 (1976), no. 3, 655–678. MR 428347, DOI https://doi.org/10.2307/2373811
- Theodore Frankel, Fixed points and torsion on Kähler manifolds, Ann. of Math. (2) 70 (1959), 1–8. MR 131883, DOI https://doi.org/10.2307/1969889
- Victor W. Guillemin and Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999. With an appendix containing two reprints by Henri Cartan [ MR0042426 (13,107e); MR0042427 (13,107f)]. MR 1689252
- V. Hauschild, Transformation groups on complex Grassmannians, to appear.
- Wu-yi Hsiang, Cohomology theory of topological transformation groups, Springer-Verlag, New York-Heidelberg, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85. MR 0423384
- Gregory Lupton and John Oprea, Cohomologically symplectic spaces: toral actions and the Gottlieb group, Trans. Amer. Math. Soc. 347 (1995), no. 1, 261–288. MR 1282893, DOI https://doi.org/10.1090/S0002-9947-1995-1282893-4
- L. N. Mann, Finite orbit structure on locally compact manifolds, Michigan Math. J. 9 (1962), 87–92. MR 132119
- Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI https://doi.org/10.2307/1970770
- Aleksy Tralle and John Oprea, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, vol. 1661, Springer-Verlag, Berlin, 1997. MR 1465676
- A. Tralle and J. Oprea, Symplectic Manifolds with no Kähler Structure, Lecture Notes in Math. 1661, Springer-Verlag, Berlin, Heidelberg, New York, 1997.
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57S15, 53D99, 55N91, 57R17
Retrieve articles in all journals with MSC (2000): 57S15, 53D99, 55N91, 57R17
Additional Information
Christopher Allday
Affiliation:
Department of Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822-2273
Email:
chris@math.hawaii.edu
Volker Hauschild
Affiliation:
Department of Mathematics, University of Calabria, I-87036 Rende, Italy
Email:
hausch@unical.it
Volker Puppe
Affiliation:
Faculty of Mathematics, University of Konstanz, D-78457 Konstanz, Germany
Email:
volker.puppe@uni-konstanz.de
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