On the topological complexity of aspherical spaces

TL;DR

This paper investigates the topological complexity of aspherical spaces, especially Eilenberg-MacLane spaces with hyperbolic fundamental groups, linking it to cohomological dimensions and essential cohomology classes.

Contribution

It establishes a connection between the topological complexity of aspherical spaces and the cohomological dimension of their fundamental groups, particularly for hyperbolic groups.

Findings

Topological complexity equals or exceeds the cohomological dimension by one for hyperbolic groups.

A spectral sequence is described to identify obstructions for cohomology classes to be essential.

The spectral sequence vanishes in the hyperbolic case, leading to the main results.

Abstract

The well-known theorem of Eilenberg and Ganea expresses the Lusternik - Schnirelmann category of an aspherical space as the cohomological dimension of its fundamental group. In this paper we study a similar problem of determining algebraically the topological complexity of the Eilenberg-MacLane spaces. One of our main results states that in the case when the fundamental group is hyperbolic in the sense of Gromov the topological complexity of an aspherical space $K(\pi, 1)$ either equals or is by one larger than the cohomological dimension of $\pi\times \pi$. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group we establish a vanishing property of this spectral sequence which leads to the main result.