Topological complexity sequences of groups

TL;DR

This paper introduces the topological complexity sequence of groups, analyzing its properties, growth, and behavior, especially for groups of infinite cohomological dimension and finite groups of even order.

Contribution

It defines a new intrinsic sequence refining topological complexity, applicable to infinite cohomological dimension groups, and studies its growth and asymptotic properties.

Findings

The sequence is weakly increasing and unbounded for infinite cohomological dimension groups.

Growth estimates and asymptotic behavior are provided for finite groups of even order.

The sequence offers a meaningful refinement of topological complexity for broader classes of groups.

Abstract

We define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike topological complexity itself, is meaningful for groups of infinite cohomological dimension. We show that the topological complexity sequence of every group of infinite cohomological dimension is weakly increasing and unbounded. We then estimate its growth and determine its asymptotic behavior for a finite group of even order.