Vertex numbers of simplicial complexes with free abelian fundamental group

Abstract

We show that the minimum number of vertices of a simplicial complex with fundamental group ${\mathbb{Z}}^n$ is at most $O(n)$ and at least $\Omega (n^{3/4})$. For the upper bound, we use a result on orthogonal 1-factorizations of $K_{2n}$. For the lower bound, we use a fractional Sylvester-Gallai result. We also prove that any group presentation $⟨S|R⟩\cong {\mathbb{Z}}^n$ whose relations are of the form $g^a{h}^b{i}^c$ for $g,h,i\in S$ has at least $\Omega (n^{3/2})$ generators.

13 pages