Borsuk - Ulam Theorems for Arbitrary S 1 Actions and Applications

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. Introduction. One of the variants of the Borsuk-Ulam Theorem states that if S2 is a bounded neighborhood of 0 in R" which is symmetric with respect to the origin and / is a continuous odd map of 3Í2 into a proper subspace of R", then/has a zero on 3Í2 [1]. An extension of this result to an infinite dimensional setting for a class of Fredholm maps was carried out by Granas [2] and more quantitative versions of the result which provide lower bounds for a topological measure of the size of/'(0) n 3S2 both in finite and infinite dimensions have been given by Holm and Spanier [3] (see also [4]). Our main goal in this paper is to obtain analogous results when fi is invariant and / equivariant with respect to an S] rather than a Z2 action. For a class of such actions which are fixed point free on R2"\{0}, available tools such as the index theory of [5] lead to an S ' version of the Borsuk-Ulam Theorem by merely repeating the arguments of [3] or [4]. However when the action is not free, this approach fails. Nevertheless we will show how a relative index theory related to the cohomological index theory of [5] can be used to obtain a Borsuk-Ulam Theorem for a class of nonfree S1 actions. In §1 some properties of this index theory will be developed in a restricted setting. (A more systematic development will be carried out in a future paper.) Some S' versions of the Borsuk-Ulam Theorem will be proved in §2. A special case is the following analogue of the Z2 result. Theorem. Let S] act linearly on R' X R2k (i.e. via a group of unitary operators) so that Fix S1 R' X {0}. Suppose that A is an annulus in R' X R2* andf: A -» R7 X R2/t', k' < k, is an equivariant map to a proper invariant subspace with f\AG —id, AG = (Fix S])í)A.Ifüisa closed, bounded invariant neighborhood of the origin with 3fi C A, ,/ie«/"'(0) n 3S2 ¥= 0, i.e., f has zeros on 3S2. Received by the editors November 15, 1981. 1980 Mathematics Subject Classification. Primary 58E05, 55B25, 57D70.