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A Hamiltonian cycle in a graph is a spanning subgraph that is homeomorphic toa circle. With this in mind, it is natural to define a Hamiltonian d-sphere ina d-dimensional simplicial complex as a spanning subcomplex that ishomeomorphic to a d-dimensional sphere.  We consider the Linial-Meshulam model for random simplicial complexes, andprove that there is a sharp threshold at $p=\sqrt{\frac{e}{\gamma n}}$ for theappearance of a Hamiltonian $2$-sphere in a random $2$-complex, where $\gamma =4^4/3^3$.

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