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arXiv · 2110.05660

Orientable triangulable manifolds are essentially quasigroups

Charlotte Aten·Semin Yoo
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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.

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Charlotte Aten, Semin Yoo. 2021-10-12. Orientable triangulable manifolds are essentially quasigroups. https://arxiv.org/abs/2110.05660

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