Basic classes and embedded spheres (Q1295230)

The author addresses the following two questions: given a smooth manifold \(X\) with a specific configuration of embedded or immersed spheres, (1) Is \(X\) of simple type or not? (2) If \(S\) is an embedded sphere with negative self-intersection, what are the possible intersection numbers \(K\cdot S\) where \(K\) is a basic class of \(X\)? After a short review of the blowup formula the author restricts his attention to the spheres with self-intersection \(p=-2\) or \(p=-3\). Then he proves the following Theorem. Let the manifold \(X\) contain an immersed sphere \(\alpha\) with one positive double point and self-intersection 0. Assume also that \(X\) contains a cohomology class \(f\in H^2 (X)\) such that \(f\cdot\alpha=1\). Then \(X\) is of simple type.