# Some new aspherical 4-manifolds

*Algebra/Geometry/Topology Seminar*

*by Hyam Rubinstein*

*Institution:*Department of Mathematics and Statistics, The University of Melbourne

*Date: Tue 19th April 2011*

*Time: 3:15 PM*

*Location: Old Arts D, The University of Melbourne*

*Abstract*: Joint work with Bell Foozwell.

In Bell's PhD thesis, he introduced the class of Haken n-manifolds, which are aspherical and in fact have universal covering by Euclidean n-space. In dimension 3, it is well-known that such manifolds are `generic' - for example, non trivial and non split knot and link complements in the 3-sphere are Haken. Thurston's geometrisation theorem proved that all such manifolds can be canonically split into pieces with metrics of non positive curvature in all except a tiny number of cases. There is a natural notion of Haken cobordisms, where two Haken n-manifolds are the boundary of a Haken n+1 manifold and the inclusion maps respect the Haken structures. A proof will be sketched that any two Haken 3-manifolds are Haken cobordant. As a corollary, this shows that any finite collection of Haken 3-manifolds can be embedded disjointly into a Haken 4-manifold so that the maps on fundamental groups are injections.

*For More Information:* contact: Craig Westerland. email: c.westerland@ms.unimelb.edu.au