Point-pushing actions for manifolds with boundary

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 $d\geq 3$ and $k\geq 1$, 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 earlier work (2021), we describe the action of the braid group of $M$ on the fibres of configuration-mapping spaces.