arXiv subjects

Başak Küçük

Publications and source records attributed to Başak Küçük.

2 recordsLinked to original sources

arXiv2025

Functoriality of the Klein-Williams Invariant and Universality Theory

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.

math.AT↗

arXiv2025

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

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.

math.AT↗