Functoriality of the Klein-Williams Invariant and Universality Theory

TL;DR

This paper investigates the functorial properties of the Klein-Williams invariant, compares it with the equivariant Lefschetz invariant, and explores their universality and relationships through explicit examples.

Contribution

It demonstrates the functoriality of the Klein-Williams invariant and develops the universality theory for such invariants, including explicit computations and comparisons.

Findings

$ ext{ell}_G(f)$ is functorial.

Explicit computation of the universal invariant group.

Comparison between $ ext{ell}_G(f)$, $ ext{lambda}_G(f)$, and the universal invariant.

Abstract

Both the Klein-Williams invariant $\ell_G(f)$ from \cite{KW2} and the generalized equivariant Lefschetz invariant $\lambda_G(f)$ from \cite{weber07} serve as complete obstructions to the fixed point problem in the equivariant setting. The latter is functorial in the sense of Definition \ref{functorial}. The first part of this paper aims to demonstrate that $\ell_G(f)$ is also functorial. The second part summarizes the ``universality" theory of such functorial invariants, developed in \cites{lueck1999, Weber06}, and explicitly computes the group in which the universal invariant lies, under a certain hypothesis. The final part explores the relationship between $\ell_G(f)$ and $\lambda_G(f)$, and presents examples to compare $\ell_G(f)$, $\lambda_G(f)$, and the universal invariant.