The prime decomposition fibre sequence for moduli spaces of 3-manifolds

Milnor's prime decomposition theorem states that every oriented 3-manifold M is diffeomorphic can be written as a connected sum of "prime" manifolds in an essentially unique way: M == P_1 # ... # P_n # (S^1 x S^2)^{#g}. This reduces many questions about 3-manifolds to the prime case, but when studying 3-manifolds in families this reduction is not so straightforward. For example, a diffeomorphism of M need not respect the decomposition into prime factors.

I will explain recent joint work with Boyd and Bregman, in which we use a homotopical version of the prime decomposition theorem to describe the classifying space BDiff(M) (the "moduli space" of M) in terms of moduli spaces of the P_i. More precisely, we establish a "prime decomposition fibre sequence" that describes the moduli space in terms of BDiff(P_1 u ... u P_n) and a space of handle-attachments that is amenable to computations. To illustrate this, I will discuss our calculation of the rational cohomology ring of BDiff((S^1 x S^2)#(S^1 x S^2)).