Algebraic & Geometric Topology 5 (2005) 1075–1109

DOI: 10.2140/agt.2005.5.1075

Abstract

If Γ is any finite graph, then the unlabelled configuration space of n points on Γ, denoted UCⁿΓ, is the space of n–element subsets of Γ. The braid group of Γ on n strands is the fundamental group of UCⁿΓ.

We apply a discrete version of Morse theory to these UCⁿΓ, for any n and any Γ, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UCⁿΓ strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Γ of degree at least 3 (and k is thus independent of n).

Keywords

graph braid groups, configuration spaces, discrete Morse theory

Mathematical Subject Classification

Primary: 20F36, 20F65

Secondary: 55R80, 57M15, 57Q05

References

Publication

Received: 26 October 2004
Accepted: 28 June 2005
Published: 31 August 2005

Authors