Orientable smooth manifolds are essentially quasigroups

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 and topics including Latin hypercubes, Johnson graphs, and Galois theory.