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Stability phenomena in the homology of tree braid groups

Eric Ramos

Algebraic & Geometric Topology 18 (2018) 2305–2337

For a tree G, we study the changing behaviors in the homology groups Hi(BnG) as n varies, where BnG := π1(UConfn(G)). We prove that the ranks of these homologies can be described by a single polynomial for all n, and construct this polynomial explicitly in terms of invariants of the tree G. To accomplish this we prove that the group nHi(BnG) can be endowed with the structure of a finitely generated graded module over an integral polynomial ring, and further prove that it naturally decomposes as a direct sum of graded shifts of squarefree monomial ideals. Following this, we spend time considering how our methods might be generalized to braid groups of arbitrary graphs, and make various conjectures in this direction.

configuration spaces of graphs, representation stability, squarefree monomial ideals
Mathematical Subject Classification 2010
Primary: 05C10
Secondary: 05E40, 05C05, 57M15
Received: 21 April 2017
Revised: 9 November 2017
Accepted: 24 January 2018
Published: 26 April 2018
Eric Ramos
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States