Back to Departmental Colloquium: Spring 2005
Departmental Colloquium
Date: Monday, Mar 21, 2005
Time: 4:15PM
Location: JWB 335
Special Colloquium Monday
How Efficiently Do 3-Manifolds Bound 4-Manifolds?
Title |
How Efficiently Do 3-Manifolds Bound 4-Manifolds? |
Abstract |
It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are many proofs of this fact, including several constructive ones, but they do not bound the complexity of the 4-manifold. (By “complexity” of a manifold we mean the minimum number of simplices in a triangulation.) Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n 2 ). It is an open question whether this quadratic bound can be replaced by a linear bound. The natural setting for this result is shadow surfaces, a representation of 3- and 4-manifolds that generalizes many other representations of these manifolds. One consequence of our results is some intriguing connections between the complexity of a shadow representation and the hyperbolic volume of a 3-manifold. Our results can also be phrased in terms of the singularities of smooth maps. In particular, the minimum number of “crossing singularities” of a map from a hyperbolic 3-manifold to the plane is bounded below and above by the hyperbolic volume. (Joint work with Francesco Costantino.) |
Representation Theory
Topology