On the Klein and Williams Conjecture for the Equivariant Fixed Point Problem

TL;DR

This paper resolves a conjecture by Klein and Williams by explicitly decomposing their invariant using the tom Dieck splitting, advancing the understanding of equivariant fixed point theory and its applications to periodic points.

Contribution

It provides an explicit decomposition of the Klein-Williams invariant, confirming that Nielsen numbers can be derived from it, and applies these results to the periodic point problem.

Findings

Explicit decomposition of Klein-Williams invariant achieved

Confirmed Nielsen numbers can be computed from the invariant

Applied results to the periodic point problem

Abstract

Klein and Williams developed an obstruction theory for the homotopical equivariant fixed point problem, which asks whether an equivariant map can be deformed, through an equivariant homotopy, into another map with no fixed points \cite[Theorem H]{KW2}. An alternative approach to this problem was given by Fadell and Wong \cite{FW88} using a collection of Nielsen numbers. It remained an open question, stated as a conjecture in \cite{KW2}, whether these Nielsen numbers could be computed from the Klein-Williams invariant. We resolve this conjecture by providing an explicit decomposition of the Klein-Williams invariant under the tom Dieck splitting. Furthermore, we apply these results to the periodic point problem.