TY - JOUR

T1 - Abrams's stable equivalence for graph braid groups

AU - Prue, Paul

AU - Scrimshaw, Travis

N1 - Publisher Copyright:
© 2014 Elsevier B.V.

PY - 2014/9/30

Y1 - 2014/9/30

N2 - In his PhD thesis [1], Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G, denoted Cn(G), deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertices of degree not equal to 2) is of length at least n+. 1 edges, and each path from a vertex to itself which is not nullhomotopic is of length at least n+. 1 edges. Using Forman's discrete Morse theory for CW-complexes, we show the first condition can be relaxed to require only that each path between distinct essential vertices is of length at least n- 1.

AB - In his PhD thesis [1], Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G, denoted Cn(G), deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertices of degree not equal to 2) is of length at least n+. 1 edges, and each path from a vertex to itself which is not nullhomotopic is of length at least n+. 1 edges. Using Forman's discrete Morse theory for CW-complexes, we show the first condition can be relaxed to require only that each path between distinct essential vertices is of length at least n- 1.

KW - Configuration space

KW - Discrete Morse theory

KW - Graph braid group

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U2 - 10.1016/j.topol.2014.09.009

DO - 10.1016/j.topol.2014.09.009

M3 - Article

AN - SCOPUS:84907540789

VL - 178

SP - 136

EP - 145

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -