Algebraic & Geometric Topology

Stability phenomena in the homology of tree braid groups

Eric Ramos

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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 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.

Article information

Algebr. Geom. Topol., Volume 18, Number 4 (2018), 2305-2337.

Received: 21 April 2017
Revised: 9 November 2017
Accepted: 24 January 2018
First available in Project Euclid: 3 May 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
Secondary: 05E40: Combinatorial aspects of commutative algebra 05C05: Trees 57M15: Relations with graph theory [See also 05Cxx]

configuration spaces of graphs representation stability squarefree monomial ideals


Ramos, Eric. Stability phenomena in the homology of tree braid groups. Algebr. Geom. Topol. 18 (2018), no. 4, 2305--2337. doi:10.2140/agt.2018.18.2305.

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