A Sharp Threshold for Spanning 2-Spheres in Random 2-Complexes

Abstract

A Hamiltonian cycle in a graph is a spanning subgraph that is homeomorphic to a circle. With this in mind, it is natural to define a Hamiltonian d-sphere in a d-dimensional simplicial complex as a spanning subcomplex that is homeomorphic to a d-dimensional sphere. We consider the Linial-Meshulam model for random simplicial complexes, and prove that there is a sharp threshold at p=eγn−−√ for the appearance of a Hamiltonian 2-sphere in a random 2-complex, where γ=44/33.