Gauge Theory and Topology Seminar: Moduli spaces of reducible 3-manifolds

The classical Kneser-Milnor theorem states that any compact, orientable 3-manifold M has a connected sum decomposition into unique irreducible prime factors P_i.  In previous work, we constructed a semi-simplicial space of all such decompositions and showed that its geometric realization is contractible. In this talk, we define a “prime decomposition map” from the classifying space BDiff^+(M) to the classifying space BDiff^+(P_M), where P_M is the disjoint union of the P_i. Moreover, we show that if M has at least one irreducible prime factor, then the homotopy fiber is finite. We also derive a spectral sequence to compute the cohomology of BDiff^+(M), and thus characteristic classes of M-bundles. This is joint work with Rachael Boyd and Jan Steinebrunner.