Geometry and Topology Seminar

Abstract: The fundamental group of a closed hyperbolic 3-manifold is known to act geometrically on a CAT(0) cube complex. We ask whether the same is true for the fundamental group of negatively curved 3-pseudomanifolds, i.e., 3-manifolds with isolated singularities. While many 3-pseudomanifolds are cubulated, such as those arising from RACGs and strict hyperbolization, in this talk we give the first examples of closed 3-pseudomanifolds that are locally CAT(-1) but whose fundamental group cannot be cubulated. These examples are obtained from certain compact hyperbolic 3-manifolds with totally geodesic boundary by coning off the boundary components. This is joint work with J. Manning.

Abstract: Let $M$ be an orientable 3-manifold.  A celebrated theorem of Kneser-Milnor states that $M$ admits a unique connected sum decomposition, up to permutations of the prime factors.  We prove a space-level version of this theorem by introducing a poset of decompositions of $M$ along collections of essential 2-spheres (called separating systems) and showing that the geometric realization of this poset is contractible.  As an application, we prove that for any $M$ the classifying space $BDiff(M)$ is homotopy equivalent to a CW complex with finite $k$-skeleton for every $k$. When M is a connected sum of $g$ copies of $S^1 \times S^2$, we also show that every topological $M$-bundle fiberwise extends to a bundle of 4-dimensional handlebodies, generalizing another classical result due to Laudenbach-Poenaru. This is joint work with Rachael Boyd and Jan Steinebrunner. 

Abstract TBA