On the topological complexity of aspherical spaces

Michael Farber; Stephan Mescher

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arXiv · 1708.06732

On the topological complexity of aspherical spaces

Michael Farber·Stephan Mescher

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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.

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Michael Farber, Stephan Mescher. 2017-08-22. On the topological complexity of aspherical spaces. https://arxiv.org/abs/1708.06732

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