Connectivity at infinity for braid groups on complete graphs

Abstract

We show that the connectivity at infinity for configuration spaces on complete graphs is determined by the connectivity of chessboard complexes.

Connectivity at Infinity for the Braid Group of a Complete Bipartite Graph

“…In [MZ13], Meier and Zhang showed that the vertex links in the configuration spaces of r robots on the complete graph K r+N are homeomorphic to ∆ r,N . If a 0-cell v of Conf r (n, N ) corresponds to a configuration in which all robots are on the side of K n,N with n vertices, then the link of this 0-cell is the same as ∆ r,N as well.…”

“…Theorem 4.7 and Lemma 4.8 go a long way towards establishing much of our argument, particularly once we show that connectivity results about vertex links quickly lead to connectivity results about punctured vertex links, as is required by Theorem 1.2. A key fact about chessboard complexes is that they are vertex decomposible, and in Section 4 of [MZ13] this fact is used to prove that connectivity is preserved when any closed simplex is removed (Theorem 4.7 of [MZ13]).…”

“…The fundamental group of this cell complex is called a graph braid group, and in this article we describe the connectivity at infinity of graph braid groups when the graph Γ is a complete bipartite graph. The case of a complete graph is settled in [MZ13].…”