2-complexes with large 2-girth
Dominic Dotterrer, Larry Guth, Matthew Kahle · arXiv (Cornell University) · 2015
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 $α 1/2$, 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.