Double plumbings of disk bundles over spheres

Abstract

We consider double plumbings of two disk bundles over spheres. We calculate the Heegaard-Floer homology with its absolute grading of the boundary of such a plumbing. Given a closed smooth 4-manifold $X$ and a suitable pair of classes in $H_{2}(X)$, we investigate when this pair of classes may be represented by a configuration of surfaces in $X$ whose regular neighbourhood is a double plumbing of disk bundles over spheres. Using similar methods we study single plumbings of two disk bundles over spheres inside $X$.