Orientable smooth manifolds are essentially quasigroups

Abstract: In my recent work with Semin Yoo we produced a generalization of a construction of Herman and Pakianathan which assigns to each finite noncommutative group a closed surface in a functorial manner. We give a pair of functors whose domain is a subcategory of a variety of n-ary quasigroups. The first of these functors assigns to each such quasigroup a smooth, flat Riemannian manifold while the second assigns to each quasigroup a topological manifold which is a subspace of the metric completion of the aforementioned Riemannian manifold. I will give examples of these constructions, draw some pictures, and argue that all homeomorphism classes of smooth orientable manifolds arise from this construction. I will then discuss a connection with the Evans Conjecture on partial Latin squares, give its implication for orientable surfaces, and state a related problem applicable to our construction for compact n-manifolds.

computational complexitycategory theorylogic

Audience: researchers in the topic

PALS Panglobal Algebra and Logic Seminar

Series comments: The PALS seminar is a research and learning seminar organized by the algebra and logic research group of the Department of Mathematics at the University of Colorado at Boulder. The scope of the seminar includes all topics with links to algebra, logic, or their applications, like general algebra, logic, model theory, category theory, set theory, set-theoretic topology, or theoretical computer science. Please contact one of the organizers for the Zoom password, to join the mailing list or if you want to speak.