Homological stability for moduli spaces of high dimensional manifolds. I

We prove a homological stability theorem for moduli spaces of simply connected manifolds of dimension 2n > 4, with respect to forming connected sum with S^n \times S^n. This is analogous to Harer's stability theorem for the homology of mapping class groups. Combined with previous work of the authors, it gives a calculation of the homology of the moduli spaces of manifolds diffeomorphic to connected sums of S^n \times S^n in a range of degrees.