Genus inequalities and four-dimensional surgery

We obtain a new genus inequality for a topologically locally flat surface in a 4-dimensional manifold. This inequality provides new information about the fundamental group of the complement of such a surface and in many cases gives the minimum genus among surfaces within the same homology class. The general problem of finding an embedded surface of a small genus allowed by the inequality remains undecided and is directly related to the surgery conjecture of 4-dimensional topology.