Dold indices and symmetric powers
M. C. Crabb
TL;DR
The paper addresses expressing Lefschetz data for symmetric-power-type constructions of a self-map in terms of iterates, extending Macdonald and Dold via a general functorial framework. It introduces Lefschetz-polynomial functors, proves their stability under additivity, multiplicativity, and composition, and applies the framework to symmetric powers, Borsuk-Ulam symmetric products, partitions, and powers, yielding explicit generating-function identities that link Lefschetz numbers to the zeta function $Z(f;q)$. Key contributions include a unified, combinatorially tractable method to compute fixed-point data from iterates, with central formulas such as $\sum_{k\ge0} \tilde{L}(S^k_l f)\,q^k = Z(f;q^{l+1}) Z(f;q)^{-1}$ and $\sum_{k\ge1} L(P^k f)\,q^k = (1-q)^{-1}(Z(f;q^2)Z(f;q)^{-1}-1)$, together with generalizations to group actions via ${}_G\Phi^K_{\mathcal{L}}$. The results unify cohomological and combinatorial viewpoints, provide tools for fixed-point analysis on configuration-type spaces, and extend to orbit-space settings, enhancing understanding of how iterates govern symmetric- and partition-type constructions. These insights advance both the theory of Lefschetz numbers and their applications to geometric topology.
Abstract
Results of Macdonald and Dold from the 1960s and '70s expressing the Lefschetz numbers of symmetric powers of a self-map of a compact ENR in terms of the Lefschetz numbers of iterates of the map are extended using the notion of a Lefschetz-polynomial functor. Configuration spaces and Borsuk-Ulam symmetric products, as well as symmetric powers, are treated as examples of the general method.