{"title":"Point-pushing actions for manifolds with boundary","authors":"Martin Palmer, U. Tillmann","doi":"10.4171/ggd/690","DOIUrl":null,"url":null,"abstract":"Given a manifold $M$ and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of $M$ to the group of isotopy classes of diffeomorphisms of $M$ that fix the basepoint. This map is well-studied in dimension $d = 2$ and is part of the Birman exact sequence. Here we study, for any $d \\geqslant 3$ and $k \\geqslant 1$, the map from the $k$-th braid group of $M$ to the group of homotopy classes of homotopy equivalences of the $k$-punctured manifold $M \\smallsetminus z$, and analyse its injectivity. Equivalently, we describe the monodromy of the universal bundle that associates to a configuration $z$ of size $k$ in $M$ its complement, the space $M \\smallsetminus z$. Furthermore, motivated by our work on the homology of configuration-mapping spaces, we describe the action of the braid group of $M$ on the fibres of configuration-mapping spaces.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2020-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Groups Geometry and Dynamics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/ggd/690","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
Given a manifold $M$ and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of $M$ to the group of isotopy classes of diffeomorphisms of $M$ that fix the basepoint. This map is well-studied in dimension $d = 2$ and is part of the Birman exact sequence. Here we study, for any $d \geqslant 3$ and $k \geqslant 1$, the map from the $k$-th braid group of $M$ to the group of homotopy classes of homotopy equivalences of the $k$-punctured manifold $M \smallsetminus z$, and analyse its injectivity. Equivalently, we describe the monodromy of the universal bundle that associates to a configuration $z$ of size $k$ in $M$ its complement, the space $M \smallsetminus z$. Furthermore, motivated by our work on the homology of configuration-mapping spaces, we describe the action of the braid group of $M$ on the fibres of configuration-mapping spaces.
期刊介绍:
Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. The journal covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields.
Topics covered include:
geometric group theory;
asymptotic group theory;
combinatorial group theory;
probabilities on groups;
computational aspects and complexity;
harmonic and functional analysis on groups, free probability;
ergodic theory of group actions;
cohomology of groups and exotic cohomologies;
groups and low-dimensional topology;
group actions on trees, buildings, rooted trees.