3:00 p.m., Friday (January 14, 2005)
Math Annex 1100
How efficiently do 3-manifolds bound 4-manifolds?
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. Given a 3-manifold M of
complexity n, we show how to construct a 4-manifold bounded by M of
complexity O (n^2), for suitable notions of "complexity". 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.
(Joint work with Francesco Costantino.)
Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).