A localization theorem for functional \(S^ 1\)-spaces
type: Item
instance of: scholarly article
title: A localization theorem for functional \(S^ 1\)-spaces
zbMATH Open document ID: 0629.57023
DOI: 10.1007/BF01460049
author: Dan Burghelea
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164477
review text: 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.
Mathematics Subject Classification ID: 57S10
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 4024323
zbMATH Keywords: S\({}^ 1\)-equivariant cohomology
zbMATH Keywords: \(S^ 1\)-space
zbMATH Keywords: fixed point set
zbMATH Keywords: Postnikov term
zbMATH Keywords: \(S^ 1\)-equivariant Euler-Poincaré characteristic
MaRDI profile type: Publication
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OpenAlex ID: W1974530369
description: scientific article; zbMATH DE number 4024323