Orientable triangulable manifolds are essentially quasigroups

TL;DR

This paper constructs a functorial method linking n-ary alternating quasigroups to smooth and topological n-manifolds, showing all connected orientable manifolds can be realized as serenations of quasigroups.

Contribution

It introduces an n-dimensional analogue of tessellated surfaces from finite groups, establishing a new connection between quasigroups and manifolds.

Findings

Every connected orientable smooth manifold is a serenation of some quasigroup.

The construction relates to Latin hypercubes and Johnson graphs.

Basic properties of the variety of alternating n-quasigroups are established.

Abstract

We introduce an $n$-dimensional analogue of the construction of tessellated surfaces from finite groups first described by Herman and Pakianathan. Our construction is functorial and associates to each $n$-ary alternating quasigroup both a smooth, flat Riemannian $n$-manifold which we dub the open serenation of the quasigroup in question, as well as a topological $n$-manifold (the serenation of the quasigroup) which is a subspace of the metric completion of the open serenation. We prove that every connected orientable smooth manifold is serene, in the sense that each such manifold is a component of the serenation of some quasigroup. We prove some basic results about the variety of alternating $n$-quasigroups and note connections between our construction, Latin hypercubes, and Johnson graphs.