TY - JOUR
T1 - Stability phenomena in the homology of tree braid groups
AU - Ramos, Eric
N1 - Publisher Copyright:
© 2018, Mathematical Sciences Publishers. All rights reserved.
PY - 2018/4/26
Y1 - 2018/4/26
N2 - 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.
AB - 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.
KW - Configuration spaces of graphs
KW - Representation stability
KW - Squarefree monomial ideals
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U2 - 10.2140/agt.2018.18.2305
DO - 10.2140/agt.2018.18.2305
M3 - Article
AN - SCOPUS:85049219777
SN - 1472-2747
VL - 18
SP - 2305
EP - 2337
JO - Algebraic and Geometric Topology
JF - Algebraic and Geometric Topology
IS - 4
ER -