A sharp threshold for spanning 2‐spheres in random 2‐complexes

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=e/γn for the appearance of a Hamiltonian 2‐sphere in a random 2‐complex, where γ=44/33.