This is the latest version of this eprint.
Torzewska, F, Goncalves Faria Martins, JN orcid.org/0000-0001-8113-3646 and Martin, P (2023) Motion Groupoids and Mapping Class Groupoids. Communications in Mathematical Physics, 402 (3). pp. 1621-1705. ISSN 0010-3616
Abstract
Here M denotes a pair (M, A) of a manifold and a subset (e.g. A = ∂ M or A = ∅). We construct for each M its motion groupoid MotM , whose object set is the power setP M of M, and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient space’ M, that fix A, acting on P M. These groupoids generalise the classical definition of a motion group associated to a manifold M and a submanifold N, which can be recovered by considering the automorphisms in MotM of N ∈ P M. We also construct the mapping class groupoid MCGM associated to a pair M with the same object class, whose morphisms are now equivalence classes of homeomorphisms of M, that fix A. We recover the classical definition of the mapping class group of a pair by taking automorphisms at the appropriate object. For each pair M we explicitly construct a functor F: MotM → MCGM , which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if π0 and π1 of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an isomorphism in the physically important case M = ([0, 1] n, ∂[0, 1] n), for any n ∈ N. We show that the congruence relation used in the construction MotM can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows— worldlines (e.g. monotonic ‘tangles’). We examine several explicit examples of MotM and MCGM demonstrating the utility of the constructions.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | © The Author(s) 2023 This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Funding Information: | Funder Grant number Leverhulme Trust RPG-2018-029 |
Depositing User: | Symplectic Publications |
Date Deposited: | 02 Jun 2023 11:49 |
Last Modified: | 08 Sep 2023 15:18 |
Status: | Published |
Publisher: | Springer Nature |
Identification Number: | 10.1007/s00220-023-04755-0 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:199500 |
Available Versions of this Item
-
Motion groupoids and mapping class groupoids. (deposited 02 Jun 2023 11:40)
- Motion Groupoids and Mapping Class Groupoids. (deposited 02 Jun 2023 11:49) [Currently Displayed]