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The random 2-complex Y=Y(n,p) is the probability space of all simplicial complexes on vertex set [n] and all edges, with each 2-dimensional face included with probability p independently. Nathan Linial and Roy Meshulam showed that if p >> 2\log{n}/n then the probability that H_{1}(Y,F_2) is trivial goes to 1 as n approaches infinity. This is an analogue of the phase transition for connectivity of the Erdos-Renyi random graph G(n,p). We show here that if p >> n^{-1/2}, then the probability that Y is simply connected goes to 1 as n approaches infinity, but if p << n^{-1/2} then the probability that Y is simply connected goes to 0. This implies in particular that vanishing of H_{1}(Y,F_2) and \pi_1(Y) have distinct thresholds. Finding the threshold for vanishing of H_{1}(Y,Z}) is still an open problem.