Stability phenomena in the homology of tree braid groups

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Abstract

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 ⊕n Hi(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.

Original languageEnglish
Pages (from-to)2305-2337
Number of pages33
JournalAlgebraic and Geometric Topology
Volume18
Issue number4
DOIs
StatePublished - 26 Apr 2018

Keywords

  • Configuration spaces of graphs
  • Representation stability
  • Squarefree monomial ideals

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