Point-pushing actions for manifolds with boundary
with Ulrike Tillmann

Groups, Geometry, and Dynamics vol. 16 no. 4 (2022) pp. 1179-1224
arXiv: abs — pdf
v3 — v2 — v1

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≥3 and k≥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∖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∖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.