Abrams's stable equivalence for graph braid groups

Details Abrams's stable equivalence for graph braid groups Abrams's stable equivalence for graph braid groups

2009-09-30

arxiv.org/abs/0909.5511v1

Mathematics

Geometric Topology

19 pages, 13 figures

Scientific paper

In his PhD thesis, Abrams proved that for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies the following two conditions. First, each path between distinct essential vertices (vertices of valence not equal to 2) is of length at least n+1 edges, and second, each path from a vertex to itself which is not nullhomotopic is of length at least n+1 edges. We prove the first condition can be relaxed to require only that each path between distinct essential vertices is of length at least n-1. In doing so, we fill a minor hole in Abrams's original proof. We show the improved result is optimal; that is, the conditions are in fact necessary for the existence of the indicated deformation retraction.

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