Motion groupoids and mapping class groupoids
Fiona Torzewska, Jo\~ao Faria Martins, Paul Purdon Martin

TL;DR
This paper introduces motion and mapping class groupoids for manifolds with subsets, generalizing classical groups, and establishes conditions under which they are isomorphic, with applications to high-dimensional cubes.
Contribution
It constructs and relates motion groupoids and mapping class groupoids for pairs of manifolds and subsets, extending classical concepts and providing explicit isomorphisms.
Findings
The functor between motion and mapping class groupoids is full and faithful under certain conditions.
The constructions recover classical groups in specific cases like the n-cube.
The congruence relation can be described via level preserving isotopy on trajectories.
Abstract
Here denotes a pair of a manifold and a subset (e.g. or ). We construct for each its motion groupoid , whose object set is the power set of , and whose morphisms are certain equivalence classes of continuous flows of the `ambient space' , that fix , acting on . These groupoids generalise the classical definition of a motion group associated to a manifold and a submanifold , which can be recovered by considering the automorphisms in of . We also construct the mapping class groupoid associated to a pair with the same object class, whose morphisms are now equivalence classes of homeomorphisms of , that fix . We recover the classical…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology
