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)



On the equivariant Morse complex of
the free loop space of a surface

Author: Nancy Hingston
Journal: Trans. Amer. Math. Soc. 350 (1998), 1129-1141
MSC (1991): Primary 58E10; Secondary 57R91, 53C22
MathSciNet review: 1458305
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove two theorems about the equivariant topology of the free loop space of a surface. The first deals with the nondegenerate case and says that the ``ordinary'' Morse complex can be given an $O(2)$-action in such a way that it carries the $O(2)$-homotopy type of the free loop space. The second says that, in terms of topology, the iterates of an isolated degenerate closed geodesic ``look like'' the continuous limit of the iterates of a finite, fixed number of nondegenerate closed geodesics.

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

  • 1. R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171-206. MR 19:859f
  • 2. M. Greenberg, Lectures on Aglebraic Topology, Benjamin, Reading MA, 1967. MR 35:6137
  • 3. D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom. 3 (1969), 493-510. MR 41:9143
  • 4. N. Hingston, An equivariant model for the free loop space of $S^{N}$, American J. of Math. 114 (1991), 139-155. MR 93b:55011
  • 5. N. Hingston, Curve shortening, equivariant Morse theory and closed geodesics on the $2$-sphere,, A.M.S. Proceedings of Symposia in Pure Mathematics, vol. 54, Amer. Math. Soc., Providence, 1993, pp. 423-429. MR 94a:00012
  • 6. N. Hingston, On the growth of the number of closed geodesics on the two-sphere, IMRN 9 (No. 4) (1993), 253-262. MR 94m:58044
  • 7. J. D. Jones, Cyclic homology and equivariant homology, Inventiones Math. 87 (1987), 403-424. MR 88f:18016
  • 8. W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag, Berlin, 1978. MR 57:17563
  • 9. W. Klingenberg, Riemannian Geometry, de Gruyter, Berlin, 1982. MR 84j:53001
  • 10. G. Lodder, Dihedral homology and the free loop space, Proc. London Math. Soc. 3 60 (1) (1990), 201-224. MR 91a:55007
  • 11. M. Morse, The Calculus of Variations in the Large, Amer. Math. Soc. Colloq. Publ., vol. 18, Amer. Math. Soc., Providence, 1934.
  • 12. H. B. Rademacher, On the equivariant Morse chain complex on the space of closed curves., Math. Z. 201 (1989), 279-302. MR 90i:58027

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58E10, 57R91, 53C22

Retrieve articles in all journals with MSC (1991): 58E10, 57R91, 53C22

Additional Information

Nancy Hingston
Affiliation: Department of Mathematics, The College of New Jersey, P. O. Box 7718, Ewing, New Jersey 08628-0718

Received by editor(s): May 4, 1996
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society