On the Klein and Williams Conjecture for the Equivariant Fixed Point Problem
Başak Küçük
TL;DR
This work resolves Klein and Williams' conjecture by decomposing the equivariant Klein–Williams invariant $\ell_G(f)$ under the tom Dieck splitting and identifying its components with reduced Reidemeister traces $\overline{R(f^H)}$. Each projection recovers Nielsen-type data, establishing that $\ell_G(f)$ vanishes if and only if $N(f^H)=0$ for all conjugacy classes $(H)$, though the individual term counts need not match unreduced Reidemeister traces. A counterexample demonstrates that $\mathcal{N}_l(f)$ can differ from $N(f^l)$ even as vanishing agrees, but the overall obstruction theory remains complete for the equivariant fixed-point problem. The paper also extends the theory to periodic points via the Fuller map, defining $\ell_n(f)$ and proving vanishing equivalences under suitable dimension conditions, with explicit connections to $N(f^k)$ for divisors $k|n$ and a concrete counterexample illustrating nuanced behavior of the projections.
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.