Point-pushing actions for manifolds with boundary (Q2694787)

Summary: 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 earlier work (2021), we describe the action of the braid group of \(M\) on the fibres of configuration-mapping spaces.