2-Complexes with Large 2-Girth

Abstract

The 2-girth of a 2-dimensional simplicial complex X is the minimum size of a non-zero 2-cycle in $H_2(X, {\mathbb {Z}}/2)$. We consider the maximum possible girth of a complex with n vertices and m 2-faces. If $m = n^{2 + \alpha }$ for $\alpha < 1/2$, then we show that the 2-girth is at most $4 n^{2 - 2 \alpha }$ and we prove the existence of complexes with 2-girth at least $c_{\alpha , \epsilon } n^{2 - 2 \alpha - \epsilon }$. On the other hand, if $\alpha > 1/2$, the 2-girth is at most $C_{\alpha }$. So there is a phase transition as $\alpha$ passes 1 / 2. Our results depend on a new upper bound for the number of combinatorial types of triangulated surfaces with v vertices and f faces.