Point-pushing actions for manifolds with boundary

TL;DR

This paper investigates the properties of point-pushing maps and braid groups in higher-dimensional manifolds, analyzing their injectivity, monodromy, and actions on configuration spaces, extending classical 2D results to higher dimensions.

Contribution

It generalizes the study of point-pushing maps and braid groups from dimension 2 to higher dimensions, analyzing their injectivity and monodromy properties.

Findings

Analysis of injectivity of the higher-dimensional point-pushing map

Description of the monodromy of the universal bundle for punctured manifolds

Action of braid groups on configuration-mapping space fibers

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.