Examples of 4-manifolds without Gompf nuclei (Q1307324)

A complex \(K3\)-surface \(X\) is a smooth simply connected 4-manifold with vanishing first Chern class (unique up to diffeomorphism) and admits a holomorphic fibration by tori over \(\mathbb C\mathbb P^1\). A Gompf nucleus \(N_2\) is defined as a regular neighbourhood of the union of a singular (or cusp) fiber and a section of such a fibration (\(X\) contains three disjoint copies of \(N_2\) corresponding to three different elliptic fibrations). A question in the Kirby list asks whether every simply connected smooth 4-manifold with \(b^+_2\geq 3\) contains a Gompf nucleus. In the present paper, the question is answered in the negative. By performing logarithmic transformations of multiplicity two on three linearly independent tori contained in the three disjoint nuclei, a 4-manifold \(X_{2,2,2}\) is obtained which does not contain \(N_2\). More generally, the existence of 4-manifolds without Gompf nuclei \(N_n\) is shown. The proofs are based on the connection between the smooth topology and the Seiberg-Witten basic classes of a given 4-manifold.