No abstract

For a graph Γ, let K(HΓ, 1) denote the Eilenberg-Mac Lane space associated to the right-angled Artin (RAA) group HΓ defined by Γ. We use the relationship between the combinatorics of Γ and the topological complexity of K(HΓ, 1) to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer n, we construct a graph On whose TC-generating function has polynomial numerator of degree n. Additionally, motivated by the fact that K(HΓ, 1) can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

For a graph $\Gamma$ , let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$ . We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$ , we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$ . Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

Using the ordered analogue of Farley-Sabalka's discrete gradient field on the configuration space of a graph, we unravel a levelwise behavior of the generators of the pure braid group on a tree. This allows us to generalize Farber's equivariant description of the homotopy type of the configuration space on a tree on two particles. The results are applied to the calculation of all the higher topological complexities of ordered configuration spaces on trees on any number of particles.

We compute the topological complexity of a polyhedral product $\mathcal{Z}$ defined {in terms of} an $\operatorname{LS}$-logarithmic family of locally compact connected $\operatorname{CW}$ topological groups. The answer is given by a combinatorial formula that involves the $\operatorname{LS}$ category of the polyhedral-product factors. As a by-product, we show that the Iwase-Sakai conjecture holds true for $\mathcal{Z}$. The proof methodology {uses} a Fadell-Husseini viewpoint for the monoidal topological complexity $\big(\mathsf{TC}^M\big)$ of a space, which, under mild conditions, recovers Iwase-Sakai's original definition. In the Fadell-Husseini context, the stasis condition - $\mathsf{TC}^M$'s \emph{raison d'\^etre} - can be encoded at the covering level. Our Fadell-Husseini inspired definition provides an alternative to the $\mathsf{TC}^M$ variant given by Dranishnikov, as well as to the ones provided by Garc\'ia-Calcines, Carrasquel-Vera and Vandembroucq in terms of relative category.