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, 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 4n 2−2α and we prove the existence of complexes with 2-girth at least c α,ǫ n 2−2α−ǫ . 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.

Minimum weight disk triangulations and fillings

“…However, the weight of this filling is actually substantially larger than W n since it contains some Cn 2 triangles with high probability. This fact was proved in [4] and further studied in the work of Dotterrer, Guth and Kahle [11] concerning the homological girth of 2-dimensional complexes which inspired our proof of Theorem 3.…”

Asymptotically good homological error correcting codes

“…We rely on the following result of Dotterer, Guth, and Kahle derived from work of Aronshtam, Linial, Łuczak and Meshulam [2]. Since a proof is omitted from [4] we sketch a proof here. For notation and details, we refer the reader to [2].…”

“…The results in [2] and [4] rely on probabilistic methods to prove the existence of such simplicial complexes. It would be interesting to have explicit constructions of such simplicial complexes.…”

“…Since the Moore bound gives a logarithmic bound on the girth of a graph in terms of the number of edges and vertices, it follows that one cannot define asymptotically good families of codes from cycle codes of graphs. By applying recent work in [4] and [2], we show in Theorem 4.2 that one can define asymptotically good families of codes from cycle codes of simplicial complexes of dimension two or higher.…”

Large Simple d-Cycles in Simplicial Complexes

“…This paper contributes to the rapidly evolving study of the combinatorics of simplicial complexes in the context of their homological and homotopical properties. Let us mention, e.g., the paper [4] dealing with small (simple) cycle in dense simplicial complexes, addressing a similar (actually, only a similar-looking) problem.…”