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 $m = n^{2 + α}$ for $α< 1/2$, then we show that the 2-girth is at most $4 n^{2 - 2 α}$ and we prove the existence of complexes with 2-girth at least $c_{α, ε} n^{2 - 2 α- ε}$. On the other hand, if $α> 1/2$, the 2-girth is at most $C_α$. So there is a phase transition as $α$ 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.

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