Homological stability for families of Coxeter groups

Research output: Contribution to journalArticle

1 Citation (Scopus)
6 Downloads (Pure)

Abstract

We prove that certain families of Coxeter groups and inclusions
W1 ,→ W2 ,→ · · · satisfy homological stability, meaning that in each degree
the homology H∗(BWn) is eventually independent of n. This gives a uniform
treatment of homological stability for the families of Coxeter groups of type
A, B and D, recovering existing results in the first two cases, and giving a
new result in the third. The key step in our proof is to show that a certain
simplicial complex with Wn-action is highly connected. To do this we show
that the barycentric subdivision is an instance of the ‘basic construction’,
and then use Davis’s description of the basic construction as an increasing
union of chambers to deduce the required connectivity.
Original languageEnglish
Pages (from-to)2779-2811
Number of pages33
JournalAlgebraic & Geometric Topology
Volume16
Issue number5
DOIs
Publication statusPublished - 7 Nov 2016

Fingerprint

Coxeter Group
Barycentric Subdivision
Homology
Deduce
Connectivity
Family
Meaning

Keywords

  • homological stability
  • Coxeter groups

Cite this

Homological stability for families of Coxeter groups. / Hepworth, Richard.

In: Algebraic & Geometric Topology, Vol. 16, No. 5, 07.11.2016, p. 2779-2811.

Research output: Contribution to journalArticle

@article{657e8c007a024bb7850ac237eba21b89,
title = "Homological stability for families of Coxeter groups",
abstract = "We prove that certain families of Coxeter groups and inclusionsW1 ,→ W2 ,→ · · · satisfy homological stability, meaning that in each degreethe homology H∗(BWn) is eventually independent of n. This gives a uniformtreatment of homological stability for the families of Coxeter groups of typeA, B and D, recovering existing results in the first two cases, and giving anew result in the third. The key step in our proof is to show that a certainsimplicial complex with Wn-action is highly connected. To do this we showthat the barycentric subdivision is an instance of the ‘basic construction’,and then use Davis’s description of the basic construction as an increasingunion of chambers to deduce the required connectivity.",
keywords = "homological stability, Coxeter groups",
author = "Richard Hepworth",
year = "2016",
month = "11",
day = "7",
doi = "10.2140/agt.2016.16.2779",
language = "English",
volume = "16",
pages = "2779--2811",
journal = "Algebraic & Geometric Topology",
issn = "1472-2747",
publisher = "Agriculture.gr",
number = "5",

}

TY - JOUR

T1 - Homological stability for families of Coxeter groups

AU - Hepworth, Richard

PY - 2016/11/7

Y1 - 2016/11/7

N2 - We prove that certain families of Coxeter groups and inclusionsW1 ,→ W2 ,→ · · · satisfy homological stability, meaning that in each degreethe homology H∗(BWn) is eventually independent of n. This gives a uniformtreatment of homological stability for the families of Coxeter groups of typeA, B and D, recovering existing results in the first two cases, and giving anew result in the third. The key step in our proof is to show that a certainsimplicial complex with Wn-action is highly connected. To do this we showthat the barycentric subdivision is an instance of the ‘basic construction’,and then use Davis’s description of the basic construction as an increasingunion of chambers to deduce the required connectivity.

AB - We prove that certain families of Coxeter groups and inclusionsW1 ,→ W2 ,→ · · · satisfy homological stability, meaning that in each degreethe homology H∗(BWn) is eventually independent of n. This gives a uniformtreatment of homological stability for the families of Coxeter groups of typeA, B and D, recovering existing results in the first two cases, and giving anew result in the third. The key step in our proof is to show that a certainsimplicial complex with Wn-action is highly connected. To do this we showthat the barycentric subdivision is an instance of the ‘basic construction’,and then use Davis’s description of the basic construction as an increasingunion of chambers to deduce the required connectivity.

KW - homological stability

KW - Coxeter groups

U2 - 10.2140/agt.2016.16.2779

DO - 10.2140/agt.2016.16.2779

M3 - Article

VL - 16

SP - 2779

EP - 2811

JO - Algebraic & Geometric Topology

JF - Algebraic & Geometric Topology

SN - 1472-2747

IS - 5

ER -