Vertex numbers of simplicial complexes with free abelian fundamental group

TL;DR

This paper investigates bounds on the minimum number of vertices needed for simplicial complexes with free abelian fundamental groups, establishing upper and lower bounds using combinatorial and algebraic techniques.

Contribution

It provides new bounds on vertex numbers for simplicial complexes with fundamental group ^n, combining combinatorial and algebraic methods to improve understanding of their structure.

Findings

Upper bound of O(n) vertices using orthogonal 1-factorizations

Lower bound of (n^{3/4}) vertices via fractional Sylvester-Gallai results

Group presentation bounds with (n^{3/2}) generators for specific relations

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 $\langle S | R\rangle \cong \mathbb{Z}^{n}$ 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.