Functoriality of the Klein-Williams Invariant and Universality Theory

Both the Klein-Williams invariant ${\mathrm{\ell }}_G(f)$ 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 ${\mathrm{\ell }}_G(f)$ 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 ${\mathrm{\ell }}_G(f)$ and ${\lambda }_G(f)$, and presents examples to compare ${\mathrm{\ell }}_G(f)$, ${\lambda }_G(f)$, and the universal invariant.