New lower bounds for the topological complexity of aspherical spaces

Abstract

Abstract. We show that the topological complexity of an aspherical space X is bounded below by the cohomological dimension of the direct product A × B, whenever A and B are subgroups of π 1 (X) whose conjugates intersect trivially. For instance, this assumption is satisfied whenever A and B are complementary subgroups of π 1 (X). This gives computable lower bounds for the topological complexity of many groups of interest (including semidirect products, pure braid groups, certain link groups, and Higman's acyclic four-generator group), which in some cases improve upon the standard lower bounds in terms of zerodivisors cup-length. Our results illustrate an intimate relationship between the topological complexity of an aspherical space and the subgroup structure of its fundamental group.

Higher topological complexity of aspherical spaces

“…Let π be a discrete group, r ≥ 2 an integer, and let K ⊂ π r = π × π · · · × π be a subgroup with the property that K ∩ H = 1 for any subgroup H ⊂ π r conjugate to the diagonal ∆ ⊂ π r . Then TC r (π) ≥ cd(K), (6) where cd(K) denotes the cohomological dimension of K. Theorem 2.1 generalises Corollary 3.5.4 from [8] (where the case r = 2 was covered) as well as the result of [18] where the class of subgroups of type K = A × B is considered assuming that r = 2.…”

“…To prove the left inequality we note that we may always take C 1 = · · · = C r−1 of size c(Γ) and C r = ∅. The estimates given by (18) are in fact the algebraic analogue of the topological estimates in (4).…”

“…is trivial. In particular, we may take K = A × B × π × · · · × π with the subgroups A, B ⊂ π satisfying A ∩ gBg −1 = {1} for any g ∈ π as in [18].…”

Parametrised topological complexity of group epimorphisms

“…Since the parametrised topological complexity of epimorphisms generalises the topological complexity of groups, we do not expect or seek a purely algebraic description. We show here however that certain bounds for the topological complexity of groups given by the author, Lupton and Oprea [16] and the author [15] admit generalizations to the parametrised setting. These bounds for TC[ρ : G ։ Q] depend only on the cohomological dimensions of various auxiliary groups, and do not require the calculation of cup products.…”

Topological complexity of unordered configuration spaces of surfaces

“…The results in this paper use a technique to bound the topological complexity of aspherical spaces developed in 2015 by Grant, Lupton and Oprea [14]. Being a homotopy invariant, the topological complexity of an aspherical space only depends on its fundamental group and the methods are algebraic in nature.…”