TY - JOUR

T1 - Edge stabilization in the homology of graph braid groups

AU - An, Byung Hee

AU - Drummond-Cole, Gabriel C.

AU - Knudsen, Ben

N1 - Publisher Copyright:
© 2020, Mathematical Sciences Publishers. All rights reserved.

PY - 2020

Y1 - 2020

N2 - We introduce a novel type of stabilization map on the configuration spaces of a graph which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains which contains strictly more information than the homology-level action. We show that the resulting differential graded module is almost never formal over the ring of edges.

AB - We introduce a novel type of stabilization map on the configuration spaces of a graph which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains which contains strictly more information than the homology-level action. We show that the resulting differential graded module is almost never formal over the ring of edges.

UR - http://www.scopus.com/inward/record.url?scp=85084433127&partnerID=8YFLogxK

U2 - 10.2140/gt.2020.24.421

DO - 10.2140/gt.2020.24.421

M3 - Article

AN - SCOPUS:85084433127

SN - 1465-3060

VL - 24

SP - 421

EP - 469

JO - Geometry and Topology

JF - Geometry and Topology

IS - 1

ER -