Discrete homotopy of token configurations

TL;DR

This paper explores the discrete homotopy theory of token configurations, establishing combinatorial analogs of classical topological results and conditions for braid groups of graphs to match their discrete versions.

Contribution

It provides a combinatorial version of Smith's theorem and conditions for isomorphism between braid groups of graphs and their discrete analogs.

Findings

Fundamental group of symmetric product is isomorphic to first homology.

Conditions identified for braid groups of graphs to be discrete analogs.

Established combinatorial versions of classical topological theorems.

Abstract

This paper studies graphical analogs of symmetric products and unordered configuration spaces in topology. We do so from the perspective of the discrete homotopy theory introduced by Barcelo et al. Our first result is a combinatorial version of a theorem of P. A. Smith, which says that the fundamental group of any nontrivial symmetric product of $X$ is isomorphic to $H_1(X)$. Our second result gives conditions under which the n-strand braid group of a graph is isomorphic to its discrete analog.