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. 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.)