{"title":"On the homology of the commutator subgroup of the pure braid group","authors":"Andrea Bianchi","doi":"10.1090/proc/15404","DOIUrl":null,"url":null,"abstract":"We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\\leq l\\leq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\\geq 2$ we have $\\mathrm{cd}([P_n,P_n])=n-2$.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"32 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2019-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/proc/15404","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\geq 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.