Stability phenomena in the homology of tree braid groups

Ramos, Eric

doi:10.2140/agt.2018.18.2305

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Stability phenomena in the homology of tree braid groups

Eric Ramos

Algebraic & Geometric Topology 18 (2018) 2305–2337

DOI: 10.2140/agt.2018.18.2305

Abstract

For a tree $G$ , we study the changing behaviors in the homology groups $H_{i} (B_{n} G)$ as $n$ varies, where $B_{n} G : = π_{1} ({UConf}_{n} (G))$ . 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 $\oplus_{n} H_{i} (B_{n} G)$ 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.

Keywords

configuration spaces of graphs, representation stability, squarefree monomial ideals

Mathematical Subject Classification 2010

Primary: 05C10

Secondary: 05E40, 05C05, 57M15

References

Publication

Received: 21 April 2017

Revised: 9 November 2017

Accepted: 24 January 2018

Published: 26 April 2018

Authors

Eric Ramos
Department of Mathematics University of Michigan Ann Arbor, MI United States

Volume 18, issue 4 (2018)