Point-pushing actions for manifolds with boundary

Given a manifold $M$ and point in its interior, the point-pushing map describes diffeomorphism that pushes along closed path. This defines homomorphism from fundamental group of to isotopy classes diffeomorphisms fix basepoint. is well-studied dimension $d=2$ part Birman exact sequence. Here we study, for any $d\geq 3$ $k\geq 1$, $k$-th braid homotopy equivalences $k$-punctured $M \smallsetminus z$, analyse injectivity. Equivalently, describe monodromy universal bundle associates configuration $z$ size $k$ complement, space $M\smallsetminus z$. Furthermore, motivated by our earlier work (2021), action on fibres configuration-mapping spaces.