Motion planning in polyhedral products of groups and a Fadell-Husseini approach to topological complexity

Abstract

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.