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[subscript 2](X,Z/2) . We consider the maximum possible girth of a complex with n vertices and m 2-faces. If m=n[superscript 2+α] for α<1/2 , then we show that the 2-girth is at most 4n[superscript 2−2α] and we prove the existence of complexes with 2-girth at least c[subscript α,ϵ]n[superscript 2−2α−ϵ]. On the other hand, if α>1/2, the 2-girth is at most Cα . 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. Keywords: Random simplicial complexes, Homology, Counting surfaces