Rational cohomology and homotopy of spaces with circle action

Abstract

This is the first of a series of papers with the aim to follow up the program initiated in [12], i.e., to investigate which classes of manifolds are symmetric in the sense that they allow some non-trivial group action. The general approach for constructing cyclic group actions on a given manifold M is to start with an action of the circle group T=S1 on some manifold in the rational homotopy type of M and to “propagate” most of the restricted cyclic group actions to M itself. In [12], we only considered free actions, whereas the focus of this and the following paper [15] is on the non-free case.This note is primarily concerned with a closer look at T-actions on Poincaré complexes and in particular with the interpretation of the Borel localization theorem in the language of deformations of algebras [14,2]. In the subsequent paper [15], we will show how to realize an abstractly given deformation in the rational homotopy category by a circle action on a manifold of the given rational homotopy type.The first two sections of this paper should be of independent interest, whereas the short final ones are mainly to be considered as background material for [15].