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. Configurations and BraidsOur 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.

The conjugacy problem in subgroups of right-angled Artin groups

“…, a N , a −1 N , in the following way: starting with the empty piling, we read the word from left to right. When a letter a i is read, we check what the last letter of the a i -stack p= a a a a 2 3 4 1 a a a a 2 3 4 1 a a a a 2 3 4 1 Figure 1. The pilings of the prefixes a −2 2 a −1 4 a 3 and a −2 2 a −1 4 a 3 a 2 , and of the full word a −2 2 a −1 4 a 3 a 2 a 4 a 1 a 2 a −1 1 a 2 2 a −1 4 .…”

“…Proof. It takes linear time to determine the graph Δ(w), and then constant time to deduce from this the generators of type (2). Now we turn to the generators of type (1), that is, the minimal roots [z i ] of the elements [w i ].…”

“…The class of groups considered in this paper contains in particular all graph braid groups [1,2,14,15,25] and more generally all state complex groups [3,16], which are closely related to robotics [2,16] (indeed, our results can be interpreted as giving very efficient algorithms for motion planning of periodic robot movements). Notice that it is still unknown whether Coxeter and Artin groups are virtually fundamental groups of compact special cube complexes [18,19], so our results do not apply immediately to these groups.…”

Topological structures of position changes in the sports

“…7) by the following binding pattern (Fig. 8), it is path connected [4]. So two athletes may be arbitrarily movable on the track of Y type.…”

Motion planning algorithm for heterogeneous combinatorial robots

“…Configuration spaces [15] were proposed to model the motion planning problems. The states of configuration are used to represent the states of the robots, obstacles, and the environment.…”