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.