Remarks on homotopy equivalence of configuration spaces of a polyhedron

Abstract

We show that the configuration space F_n(M) of n particles in a compact connected PL manifold M with nonempty boundary partial M is homotopy equivalent to the configuration space F_n({mathrm{Int}}, M) where . Actually we prove some generalization of this result for polyhedra. Similar results recently have been obtained independently for topological manifolds by Zapata (Collision-free motion planning on manifolds with boundary, 2017. arXiv:1710.00293), using different techniques. We also address the question of whether a compact PL manifold M can be approximated up to homotopy type by discrete configuration spaces defined combinatorially via a simplicial subdivision of M.