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. \vskip .1in \noindent 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. \vskip .1in \noindent (joint work with francesco costantino.)