The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

Abstract: This is a paper in smooth $4$-manifold topology, inspired by the Mordell-Weil Theorem in number theory. More precisely, we prove a smooth version of the Mordell-Weil Theorem and apply it to the `unipotent radical' case of a Thurston-type classification of mapping classes of simply-connected $4$-manifolds $M_d$ that admit the structure of an elliptic complex surface of arithmetic genus $d\geq 1$. Applications include Nielsen realization theorems for $M_d$. By combining this with known results, we obtain the following remarkable consequence: if the singular fibers of such an elliptic fibration are of the simplest (i.e.\ nodal) type, then the fibered structure is unique up topological isotopy. In particular, any diffeomorphism of $M_d,d\geq 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers.