Tensor rank of the determinant and periodic triangulations of ${\mathbb{R}}^n$

We prove that in any ${\mathbb{Z}}^n$-periodic triangulation of ${\mathbb{R}}^n$ the number of ${\mathbb{Z}}^n$-orbits of $n$-dimensional simplices is at least the tensor rank of the $n$th determinant tensor. The latter is known to be at least $\frac{n^{n-1}}{(n-1)!}$, which is approximately $\frac{e^n}{\sqrt{2\pi n}}$ for large $n$. The triangulation is not assumed to be geometric, meaning that its simplices can be ``curved’'. We also provide lower bounds for general spaces. A simplicial cell complex is a CW-complex glued out of simplices with the attaching maps being simplicial embeddings; this notion generalizes simplicial complexes. We prove that if $X$ is a simplicial cell complex with cohomological classes ${\alpha }_i\in H^{d_i}(X;{\mathbb{Z}}_2)$ satisfying $${\alpha }_1⌣{\alpha }_2⌣\dots ⌣{\alpha }_n\mathrm{\ne }0,$$ then $X$ has at least $2^n$ simplices of dimension $d_1+d_2+\dots +d_n$. In particular, a simplicial cell complex homeomorphic to $\mathbb{R}{P}^n$, $\mathbb{C}{P}^n$, or $(S^2{)}^n$, has at least $2^n$ top-dimensional simplices. A crystallization of a manifold is a simplicial cell complex homeomorphic to this manifold and having the least possible number of vertices. We give a short explicit construction of a crystallization and a triangulation of ${\mathbb{R}}^n\mathrm{/}{\mathbb{Z}}^n$ with $n+1$ and $2^{n+1}-1$ vertices, resp. Triangulations with this many vertices were described before and no smaller triangulation is known.