A problem of Kirby (Problem 4.98 in ) will be answered in the negative. We show that the 4-manifold X2,3,2 defined below does not contain the Gompf nucleus N2. More generally, we also show the existence of 4-manifolds without Gompf nuclei Nn. The proofs rely on the connection between the smooth topology and the Seiberg-Witten basic classes of a given 4-dimensional manifold M.
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