The homotopy fixed point set of Lie group actions on elliptic spaces

TL;DR

This paper investigates the homotopy fixed point set of Lie group actions on elliptic spaces, showing each component remains elliptic and providing algebraic models to analyze the inclusion of fixed points.

Contribution

It introduces an explicit algebraic model for the inclusion of fixed points and analyzes the homotopy type of fixed point sets for elliptic spaces under Lie group actions.

Findings

Each component of the homotopy fixed point set is elliptic.

The algebraic model allows analysis of the inclusion map.

For torus actions, the induced map on rational homotopy groups is injective.

Abstract

Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$-space. In this paper we analyze the homotopy type of the homotopy fixed point set $X^{hG}$, and the natural injection $k\colon X^G\hookrightarrow X^{hG}$. We show that if $X$ is elliptic, that is, it has finite dimensional rational homotopy and cohomology, then each path component of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the inclusion $k$ based on which we can prove, for instance, that for $G$ a torus, $\pi_*(k)$ is injective in rational homotopy but, often, far from being a rational homotopy equivalence.