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

Abstract

An S] version of the Borsuk-Ulam Theorem is proved for a situation where Fix S1 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' symmetry.