Motion planning in polyhedral products of groups and a Fadell-Husseini approach to topological complexity - Aguilar-Guzmán, Jorge Aguilar-Guzmán; González, Jesús (Topological Methods in Nonlinear Analysis, 2023)
We compute the topological complexity of a polyhedral product Z defined in terms of an LS-logarithmic family of locally compact connected CW topological groups. The answer is given by a combinatorial formula that involves the LS category of the polyhedral-product factors. As a by-product, we show that the Iwase-Sakai conjecture holds true for Z. The proof methodology uses a Fadell-Husseini viewpoint for the monoidal topological complexity (TC^M) of a space, which, under mild conditions, recovers Iwase-Sakai’s original definition. In the Fadell-Husseini context, the stasis condition - TC^M’s raison d’être - can be encoded at the covering level. Our Fadell-Husseini inspired definition provides an alternative to the 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.