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 earlier work (2021), we describe the action of the braid group of M on the fibres of configuration-mapping spaces.
1
Academiei Române, București, Romania
2
University of Oxford, UK
Martin Palmer; Ulrike Tillmann. Point-pushing actions for manifolds with boundary. Groups, geometry, and dynamics, Tome 16 (2022) no. 4, pp. 1179-1224. doi: 10.4171/ggd/690
@article{10_4171_ggd_690,
author = {Martin Palmer and Ulrike Tillmann},
title = {Point-pushing actions for manifolds with boundary},
journal = {Groups, geometry, and dynamics},
pages = {1179--1224},
year = {2022},
volume = {16},
number = {4},
doi = {10.4171/ggd/690},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/690/}
}
TY - JOUR
AU - Martin Palmer
AU - Ulrike Tillmann
TI - Point-pushing actions for manifolds with boundary
JO - Groups, geometry, and dynamics
PY - 2022
SP - 1179
EP - 1224
VL - 16
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/690/
DO - 10.4171/ggd/690
ID - 10_4171_ggd_690
ER -
%0 Journal Article
%A Martin Palmer
%A Ulrike Tillmann
%T Point-pushing actions for manifolds with boundary
%J Groups, geometry, and dynamics
%D 2022
%P 1179-1224
%V 16
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/690/
%R 10.4171/ggd/690
%F 10_4171_ggd_690
