2-complexes with large 2-girth

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.