An elementary proof of the chromatic Smith fixed point theorem

A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if $A$ is a finite abelian $p$-group of rank $r$, then any finite $A$-space $X$ which is acyclic in the $n$th Morava $K$-theory with $n \geqslant r$ will have its subspace $X^A$ of fixed points acyclic in the $(n-r)$th Morava Ktheory. This is a chromatic homotopy version of P. A. Smith’s classical theorem that if $X$ is acyclic in mod p homology, then so is $X^A$.

The main purpose of this paper is to give an elementary proof of this new theorem that uses minimal background, and follows, as much as possible, the reasoning in standard proofs of the classical theorem. We also give a new fixed point theorem for finite dimensional, but possibly infinite, $A\textrm{-CW}$ complexes, which suggests some open problems.

Keywords

fixed point, chromatic homotopy, Morava $K$-theory

2010 Mathematics Subject Classification

Primary 55M35. Secondary 55N20, 55P42, 55P91.

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Received 3 March 2023

Accepted 24 March 2023

Published 21 February 2024