Finding Topology in a Factory: Configuration Spaces

Abstract

It is perhaps not universally acknowledged that an outstanding place to find interesting topological objects is within the walls of an automated warehouse or factory. The examples of topological spaces constructed in this exposition arose simultaneously from two seemingly disparate fields: the first author, in his thesis [1], discovered these spaces after working with the group of H. Landau, Z. Landau, J. Pommersheim, and E. Zaslow [2] on problems about multiple random walks on graphs. The second author [8, 7] discovered these same spaces while collaborating with D. Koditschek in the Artificial Intelligence Lab at the University of Michigan. The net result makes evident the abundance of topological objects within the physical world. Topology seeks to describe, as one author puts it, the "shape of space" [15], with "shape" being interpreted as appropriate for the context at hand. We will begin with thinking about spaces up to homeomorphism (continuous maps with continuous inverse), but will quickly need to abandon this class in favor of a looser form of equivalence: homotopy type. Although few topological prerequisites are necessary for fully appreciating the examples discussed here, the class of spaces we consider gives an earthly incarnation of several intricate ideas from topology, such as K(π, 1) spaces (a.k.a. Eilenberg-MacLane spaces of type K(π, 1)) and NPC (or non-positively curved) spaces. 1. Configurations and Braids Our story begins with a classical construction: that of a configuration space of points. We consider first the configuration space of N distinct labelled points RG supported in part by NSF Grant # DMS-9971629. The authors wish to thank Margaret Symington for her careful reading of the manuscript.

Graph braid groups and right-angled Artin groups

“…is an exterior face algebra of a non-flag complex, the induced map i * is degree-preserving and ker(i * ) is generated by homogeneous elements of degree 1 and 2, then H * (B n Γ; Z 2 ) is not an exterior face algebra of a flag complex by Proposition 3.1 (2). So B n Γ is not a right-angled Artin group by Proposition 3.1(1).…”

“…Each cell c occurring in R(c) also contains at least one of e and e s . Conditions (1) and (2) guarantee that e remains order-respecting and satisfies (1) and (2). If c contains e s and every vertex v in c with τ (e s ) < v < ι(e s ) is blocked, then c is collapsible.…”

“…In fact, all graph braid groups were presumed to be right-angled Artin groups until it was shown by Abrams and Ghrist in [2] that the pure 2-braid groups of the complete graph K 5 and the complete bipartite graph K 3,3 are surface groups, so they are not right-angled Artin groups. Since then, it was a reasonable conjecture that every planar-graph braid group is a right-angled Artin group.…”

An introduction to right-angled Artin groups

“…Right-angled Artin groups also appear in the study of certain problems in robotics. Abrams and Ghrist (see eg., [1] , [2], [54]) consider the case of robots moving along tracks on a factory floor. Since two robots cannot occupy the same space at the same time, the possible positions of the robots is the configuration space of n distinct points on the graph formed by the tracks.…”

Connectivity at infinity for braid groups on complete graphs

“…The recent interest in braid groups on graphs was sparked by work of Abrams and Ghrist [1,2,13]. In addition to highlighting the connection to robot motion planning, these papers present non-positively curved classifying spaces for these groups.…”