Proves dTC(Γ)=TC(Γ) for torsion-free hyperbolic and nilpotent groups, shows dTC(L^n_p)≤2p-1 and dcat(L^n_p)≤p-1 (equality in some cases), and derives counterexamples to product formulas.
On the topological complexity of aspherical spaces
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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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2025 1verdicts
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On distributional topological complexity of groups and manifolds
Proves dTC(Γ)=TC(Γ) for torsion-free hyperbolic and nilpotent groups, shows dTC(L^n_p)≤2p-1 and dcat(L^n_p)≤p-1 (equality in some cases), and derives counterexamples to product formulas.