Degree-one maps, surgery and four-manifolds

We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold M to a closed, oriented 3-manifold N if and only if M can be obtained from N by surgery about a link in N each of whose components is an unknot. We use this to interpret the existence of degree-one maps between closed 3-manifolds in terms of smooth 4-manifolds. More precisely, we show that there is a degree-one map from M to N if and only if there is a smooth embedding of M in $W = (N \times I)\#_n\bar{CP^2}\#_mCP^2$, for some $m\ge 0$, $n\ge 0$ which separates the boundary components of W. This is motivated by the relation to topological field theories, in particular the invariants of Ozsvath and Szabo