  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.
