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