  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 $p=\sqrt{\frac{e}{\gamma n}}$ for the
appearance of a Hamiltonian $2$-sphere in a random $2$-complex, where $\gamma =
4^4/3^3$.
