Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The fixed point theorem in equivariant cohomology

Authors: J. D. S. Jones and S. B. Petrack
Journal: Trans. Amer. Math. Soc. 322 (1990), 35-49
MSC: Primary 58A10; Secondary 55N35, 55N91, 58A12
MathSciNet review: 1010411
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the $ {S^1}$-equivariant de Rham cohomology of infinite dimensional $ {S^1}$-manifolds. Our main example is the free loop space $ LX$ where $ X$ is a finite dimensional manifold with the circle acting by rotating loops. We construct a new form of equivariant cohomology $ h_T^*$ which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces $ LX$. This gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of Witten [16], Atiyah [2], and Bismut [5]. In particular we show, following Atiyah in [2], that the $ \hat A$-polynomial of $ X$ arises as an equivariant characteristic class, in the theory $ h_T^*$, of the normal bundle to $ X$, considered as the space of constant loops, in $ LX$.

References [Enhancements On Off] (What's this?)

  • [1] R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, tensor analysis, and applications, Addison-Wesley, 1983. MR 697563 (84h:58001)
  • [2] M. F. Atiyah, Circular symmetry and stationary phase approximation, Colloque en I'honneur de Laurent Schwartz, Astérisque 131 (1985), 311-323. MR 816738 (87h:58206)
  • [3] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. MR 721448 (85e:58041)
  • [4] N. Berline and M. Vergne, Zéros d'un champ des vecteurs et classes charactéristiques équivariantes, Duke Math. J. 50 (1983), 539-548. MR 705039 (84i:58114)
  • [5] J.-M. Bismut, Index theorem and equivariant cohomology on the loop space, Comm. Math. Phys. 98 (1985), 213-237. MR 786574 (86h:58129)
  • [6] -, Localisation formulas, superconnections and the index theorem for families, Comm. Math. Phys. 103 (1986), 127-166. MR 826861 (87f:58147)
  • [7] J. J. Duistermatt and G. J. Heckmann, On the variation in the cohomology of the symplectic form on the reduced phase space, Invent. Math. 93 (1981), 139-149.
  • [8] E. Getzler, J. D. S. Jones and S. B. Petrack, Cyclic homology, loop spaces and the Chern character, Operator Algebras and Applications, Vol. 1, edited by D. E. Evans and M. Takesaki, LMS Lecture Note Series 135, Cambridge Univ. Press, 1988, pp. 95-108. MR 996442 (90g:58124)
  • [9] E. Getzler, J. D. S. Jones and S. B. Petrack, Differential forms on loop spaces and the cyclic bar construction, Topology (to appear).
  • [10] T. G. Goodwillie, Cyclic homology, derivations and the free loop space, Topology 24 (1985), 187-217. MR 793184 (87c:18009)
  • [11] J. D. S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87, 403-423. MR 870737 (88f:18016)
  • [12] J. D. S. Jones and S. B. Petrack, Le théorème des points fixes en cohomologie équivariante en dimension infinie, C.R. Acad. Sci. Paris Ser. I 306 (1988), 75-78. MR 929113 (89b:58012)
  • [13] J. Milnor and J. Stasheff, Characteristic classes, Ann. of Math. Studies, no. 76, Princeton Univ. Press, Princeton, N. J., 1974. MR 0440554 (55:13428)
  • [14] V. Matthai and D. G. Quillen, Superconnections, equivariant differential forms and the Thom class, Topology 25 (1986), 85-110. MR 836726 (87k:58006)
  • [15] C. Taubes, On the cohomology of elliptic genera, Preprint, Harvard University, 1987.
  • [16] E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692. MR 683171 (84b:58111)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58A10, 55N35, 55N91, 58A12

Retrieve articles in all journals with MSC: 58A10, 55N35, 55N91, 58A12

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society