Double plumbings of disk bundles over spheres (Q2515003)

Let \(N_{m,n}\) be the double plumbing of two disk bundles with Euler classes \(m\) and \(n\) over spheres, and let \(Y_{m,n}\) be the boundary of \(N_{m,n}\). The main technical result (Theorem 1.1) evaluates Heegaard--Floer homology of \(Y_{m,n}\) with respect to certain Spin\(^c\) structures on \(Y\). As applications, the author discuss if \(N_{m,n}\) can occur inside a 4-manifold \(X\) with \(H_2^+(X)=2\). Here we have the following results. Theorem 1.2(a): Any two spheres representing classes \((2,2), (2,-1)\in H_2(\mathbb CP^2\# \mathbb CP^2)\) intersect with at least 4 geometric intersections, and there exist representatives with exactly 4 intersections. Theorem 1.3: For all \(k>1\), any two spheres representing classes \((2k+1,2,0,0), (-k,1,2k,1)\in H_2(S^2\times S^2\# S^2\times S^2)\) intersect with at least 3 geometric intersections.