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)



Borsuk-Ulam theorems for arbitrary $ S\sp{1}$ actions and applications

Authors: E. R. Fadell, S. Y. Husseini and P. H. Rabinowitz
Journal: Trans. Amer. Math. Soc. 274 (1982), 345-360
MSC: Primary 55M20; Secondary 58E05
MathSciNet review: 670937
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An $ {S^1}$ version of the Borsuk-Ulam Theorem is proved for a situation where Fix $ {S^1}$ may be nontrivial. The proof is accomplished with the aid of a new relative index theory. Applications are given to intersection theorems and the existence of multiple critical points is established for a class of functional invariant under an $ {S^1}$ symmetry.

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

  • [1] J. T. Schwartz, Nonlinear functional analysis, Gordon & Breach, New York, 1969. MR 0433481 (55:6457)
  • [2] A. Granas, The theory of compact vector fields and some of its applications to the topology of function spaces. I, Rozprawy Mat. Warsaw 30 (1962). MR 0149253 (26:6743)
  • [3] P. Holm and E. H. Spanier, Involutions and Fredholm maps, Topology 10 (1971), 203-218. MR 0290410 (44:7591)
  • [4] P. H. Rabinowitz, A note on a nonlinear elliptic equation, Indiana Univ. Math. J. 22 (1972), 43-49. MR 0338554 (49:3318)
  • [5] E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139-174. MR 0478189 (57:17677)
  • [6] P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. (to appear). MR 662065 (83k:35037)
  • [7] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. MR 0370183 (51:6412)
  • [8] E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [9] V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. (to appear). MR 675067 (84c:58014)
  • [10] -, A geometrical index for the group $ {S^1}$ for some applications to the research of periodic solutions of O. D. E.'s, Comm. Pure Appl. Math. (to appear).
  • [11] A. Bahri, Une méthode perturbative en théorie de Morse, preprint.
  • [12] D. C. Clark, A variant of the Ljusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972), 65-74. MR 0296777 (45:5836)
  • [13] P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, Proc. Sympos. on Eigenvalues of Nonlinear Problems, Edizioni Cremonese, Rome, 1974, pp. 141-195. MR 0464299 (57:4232)
  • [14] I. Ekeland and J. M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. (to appear). MR 592293 (81m:58032)
  • [15] G. E. Bredon, Introduction to compact transformation groups, Academic Press, London and New York, 1972. MR 0413144 (54:1265)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55M20, 58E05

Retrieve articles in all journals with MSC: 55M20, 58E05

Additional Information

Keywords: Equivariant cohomology, index theory, Borsuk-Ulam Theorem, intersection theorems, minimax, critical point, Hamiltonian system
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society