Note on Mapping Class Groups of Finite Spaces
B. Branman

TL;DR
This paper studies the mapping class groups of finite and certain non-Hausdorff spaces, revealing their structure and showing every finite group can be realized as such a group.
Contribution
It establishes an isomorphism between the mapping class group of a finite space and the homeomorphism group of its $T_0$ quotient, and demonstrates that every finite group arises as a mapping class group.
Findings
Mapping class group of a finite space is isomorphic to the homeomorphism group of its $T_0$ quotient.
Every finite group can be realized as the mapping class group of some finite, $T_0$ space.
Abstract
We investigate the mapping class groups of a class of non-Hausdorff topological spaces which includes finite spaces. We show that the mapping class group of a finite space is isomorphic to the homeomorphism group of its quotient. As a corollary, we show that every finite group is the mapping class group of some finite, space.
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
TopicsFinite Group Theory Research · Coding theory and cryptography · Advanced Topology and Set Theory
