School Seminars and Colloquia

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.

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