Title: On the topological complexity of aspherical spaces,
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.