Representing homology classes by locally flat surfaces of minimum genus

A necessary and sufficient condition is given for a nontrivial homology class of a simply connected 4-manifold to be represented by a simple, topologically locally flat embedding of a compact Riemann surface. This result extends the embedding theorem of the 2-sphere, proved by the authors in K-Theory7 (1993), to the case of a surface of any genus. As a consequence, the computation of the least genus of a locally flat surface is obtained for any homology class of even divisibility. The computation of the least genus is also settled for certain (and perhaps all) classes of odd divisibility. In the case of a minimum genus surface it is shown that the constructed embedding is often nonsmoothable. For example, all but nine homology classes in the complex projective plane contain such a nonsmoothable surface. A similar result holds for any class of nonnegative self intersection in a K3 surface.