It has been 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, but little attention has been paid to the complexity of the 4-manifold produced. Given a 3-manifold M 3 of complexity n, we construct a 4-manifold bounded by M of complexity , where the ‘complexity’ of a piecewise-linear manifold is the minimum number of n-simplices in a triangulation.The proof goes through the notion of… CONTINUE READING