Homology classes of negative square and embedded surfaces in 4-manifolds (Q2865916)

A well-known problem in 4-manifold theory is to determine the minimal genus of surfaces that represent a given 2-homology class. A related problem in Kirby's problem list (Problem 4.105) asks whether there is a lower bound on self-intersection numbers of embedded spheres in a given smooth 4-manifold. More generally, one can ask whether there is a lower bound on the self-intersection numbers of embedded surfaces of an arbitrary genus. The problem is harder and less is known in the case of homology classes of negative self-intersection. Let \(M\) be a simply connected 4-manifold. The author proves that there is a negative lower bound on the self-intersection numbers for embedded surfaces for a given homology class that represents either a divisible class or a characteristic class. Here a two-dimensional homology class in a 4-manifold is called characteristic if its intersection pairing with any integral degree 2 homology class \(A\) is equal mod 2 to the self-intersection of \(A\).