ar X iv : m at h / 05 06 57 7 v 1 [ m at h . G T ] 2 8 Ju n 20 05 3 - MANIFOLDS EFFICIENTLY 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 several proofs of this fact, including constructive proofs, but there has been little attention to the complexity of the 4-manifold produced. Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n). Here we measure “complexity” of a piecewise-linear manifold by the minimum number of nsimplices in a triangulation. It is an open question whether this quadratic bound can be replaced by a linear bound. The proof goes through the notion of “shadow complexity” of a 3-manifold M . A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M . We prove that, for a manifold M satisfying the Geometrization Conjecture with Gromov norm G and shadow complexity S,