Jan Steinebrunner: Moduli spaces of 3-manifolds with boundary are finite

Abstract.

 In joint work with Rachael Boyd and Corey Bregman we study the classifying space \(\operatorname{BDiff}(M)\) of the diffeomorphism group of a connected, compact, orientable 3-manifold \(M\). I will recall this \(\operatorname{BDiff}(M)\), which is also called the "moduli space of \(M\)", and explain how it parametrises smooth families of manifolds diffeomorphic to \(M\).
 
Using Milnor's prime decomposition and Thurston's geometrisation conjecture we can cut \(M\) into "geometric pieces", for which we have a better understanding of \(\operatorname{BDiff}\). The purpose of this talk is to explain a technique for computing the moduli space \(\operatorname{BDiff}(M)\) in terms of the moduli spaces of the pieces. We use this to prove that if \(M\) has non-empty boundary, then \(\operatorname{BDiff}(M \mathbin{\text{rel}} \partial)\) has the homotopy type of a finite CW complex, as was conjectured by Kontsevich.