UC Berkeley

Given a smooth, closed, oriented 4-manifold X and alpha is an element ofH(2)(X, Z) such that alpha . alpha > 0, a closed 2-form omega is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles Z. The number of circles, counted with sign, is given by d = (c(1)(s)(2)-3sigma(X)-2chi(X))/4, where s is a certain spin(C) structure naturally associated to omega.