{"title":"Homological stability and densities of generalized configuration spaces","authors":"Q. Ho","doi":"10.2140/gt.2021.25.813","DOIUrl":null,"url":null,"abstract":"We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types, answering a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.","PeriodicalId":55105,"journal":{"name":"Geometry & Topology","volume":"9 1","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2018-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometry & Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/gt.2021.25.813","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 6
Abstract
We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types, answering a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.
期刊介绍:
Geometry and Topology is a fully refereed journal covering all of geometry and topology, broadly understood. G&T is published in electronic and print formats by Mathematical Sciences Publishers.
The purpose of Geometry & Topology is the advancement of mathematics. Editors evaluate submitted papers strictly on the basis of scientific merit, without regard to authors" nationality, country of residence, institutional affiliation, sex, ethnic origin, or political views.