Geometric Analysis and Topology Seminar

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On diffeomorphism groups of reducible 3-manifolds

Speaker: Corey Bregman, Tufts University

Location: Warren Weaver Hall 512

Date: Friday, October 24, 2025, 11 a.m.

Synopsis:

Let M be a smooth, compact, orientable 3-manifold. A celebrated result of Kneser and Milnor states that M admits a connected sum decomposition into prime factors, unique up to reordering. In general, however, there are infinitely many isotopy classes of spheres in M which yield such a decomposition.  We introduce a topological poset of embedded 2-spheres in M and use it to study the classifying space BDiff(M) for the diffeomorphism group of M.  We prove that if M is closed then BDiff(M) has finite type, and if M has non-empty boundary then BDiff(M rel ∂M) is homotopy equivalent to a finite CW complex.  The theory we develop has other applications for which I’ll provide a brief overview. This is joint work with Rachael Boyd and Jan Steinebrunner.