Chor Hang Lam

To study the homology of classifying spaces of diffeomorphism groups of compact, orientable 3-manifolds, we use the stabilization map that glues a prime 3-manifold, P , to an existing manifold N with non-empty boundary via connected sum. The main result is that the stabilization map induces a homology isomorphism for the i-th homology after the number n of existing P summands in N is greater than 2i + 2. We also examine stabilization by the 3-ball, D3 , for the case P = S 1 x S 2 , induces homology isomorphism in the same range as before, i.e. if the number n of existing P summands in N is greater than 2i + 2. This result allows us to exhibit homological stability even if N does not have boundary in the case P = S 1 x S 2 . Along with known results about the stable homology, this gives a way to compute homology of B Diff(#n S 1 x S 2 ) in the range where homological stability holds.