A localization theorem for functional \(S^ 1\)-spaces
The \(S^ 1\)-equivariant cohomology \(H_ *^{S^ 1}(\tilde X;Q)\) of an \(S^ 1\)-space \~X, is a module over \(H_ *^{S^ 1}(pt;Q)=Q[u]\), deg u\(=2\). Given a compact \(S^ 1\)-manifold \~K (possibly with boundary) and an \(S^ 1\)-space \~X, let Maps(\~K,\~X) be the space of continuous maps from K to X equipped with the diagonal action. In this paper we study \(H^ *_{S^ 1}(Maps(\tilde K,\tilde X);Q)\) up to Q[u]-torsion. The main result shows that it depends only on Maps(\=F,\~X) and Maps(\~K,\~X(dim K)) where \=F is a fixed point set of \~K (viewed as an \(S^ 1\)-space with trivial action and \~X(r) the r-Postnikov term of \~X. As application we give results about the \(S^ 1\)-equivariant Euler- Poincaré characteristic (of function spaces), a peculiar extension of the usual Poincaré characteristic.
\(S^ 1\)-equivariant Euler-Poincaré characteristic
scientific article; zbMATH DE number 4024323