Functoriality of the Klein-Williams Invariant and Universality Theory
Başak Küçük
TL;DR
The paper proves that the Klein–Williams invariant $\ell_G(f)$ is a functorial equivariant Lefschetz invariant and situates it within the universal theory of functorial equivariant Lefschetz invariants. It provides explicit computations of the universal invariant $U_G^{\mathbb{Z}}(X,f)$ in the simply-connected, non-equivariant case (showing $U^{\mathbb{Z}}(X,f) \cong U(\mathbb{Z})$) and solves the realization problem there. It then relates $\ell_G(f)$ to Weber’s generalized invariant $\lambda_G(f)$ via a trace map, proving that vanishing of $\ell_G(f)$ and $\lambda_G(f)$ coincide under the gap hypothesis, while also demonstrating with examples that the two invariants need not coincide in general. Collectively, the work clarifies how these invariants capture fixed-point obstructions in both non-equivariant and equivariant settings and illustrates their distinct algebraic and geometric content through concrete computations.
Abstract
Both the Klein-Williams invariant $\ell_G(f)$ from \cite{KW2} and the generalized equivariant Lefschetz invariant $λ_G(f)$ 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 $λ_G(f)$, and presents examples to compare $\ell_G(f)$, $λ_G(f)$, and the universal invariant.