Given a manifold 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 to the group of isotopy classes of diffeomorphisms of that fix the basepoint. This map is well-studied in dimension and is part of the Birman exact sequence. Here we study, for any and , the map from the -th braid group of to the group of homotopy classes of homotopy equivalences of the -punctured manifold , and analyse its injectivity. Equivalently, we describe the monodromy of the universal bundle that associates to a configuration of size in its complement, the space . Furthermore, motivated by our earlier work (2021), we describe the action of the braid group of on the fibres of configuration-mapping spaces.
Cite this article
Martin Palmer, Ulrike Tillmann, Point-pushing actions for manifolds with boundary. Groups Geom. Dyn. 16 (2022), no. 4, pp. 1179–1224DOI 10.4171/GGD/690