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 $4n^{2-2\alpha }$ 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.

mostly minor revisions from previous version