Bounds on Genus and Geometric Intersections from Cylindrical End Moduli Spaces

In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive self-intersection in a smooth four-manifold X with second positive Betti number b+2 (X) = 1. We study the solutions of the Seiberg-Witten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this approach to derive lower bounds on the number of intersection points of n algebraically disjoint surfaces of positive self-intersection in manifolds with b 2 (X) = n.