Sectional category of subgroup inclusions and sequential topological complexities of aspherical spaces as A-genus
Arturo Espinosa Baro
TL;DR
This work establishes a precise link between subgroup-based sectional categories and the r-th sequential topological complexity of aspherical spaces through the A-genus, a robust equivariant invariant. It proves that for torsion-free groups G and subgroups H, the sectional category of the inclusion H→G is governed by the A-genus of EG with respect to the single-family {G/H}, and, for diagonally embedded subgroups in G^r, the r-th TC of a K(π,1) space coincides with the A-genus of the universal cover, with sharp lower bounds provided by the A-genus in general and equalities in the aspherical case. The authors derive a suite of computable bounds for secat and TC_r in terms of classifying spaces for families and Bredon cohomological dimensions, recovering known results (e.g., FOSequ) and yielding new bounds for semidirect products and diagonal subgroups. Beyond the classical setting, they introduce and explore proper genus and underline TC(G) to study topological complexity under proper group actions, including explicit examples (e.g., pmm and p3) and qualitative gaps between proper genus and proper TC, and they discuss A-nullification as a tool to model underline B G in terms of BG. The work thus provides a unifying, computable framework connecting group-theoretic invariants with topological complexity in both ordinary and proper-action contexts, with potential impact on category-like invariants for groups and their actions.
Abstract
In this paper we characterize the sectional category of subgroup inclusions and the $r^{th}$-sequential topological complexity of aspherical spaces of a group G in terms of the A-genus in the sense of Clapp-Puppe and Bartsch for a suitable one-element family of G-spaces A, and we discuss some of the consequences of such characterization, including new ideas about notions of category-like invariants with respect to proper actions of groups.