On dissolving knot surgery 4-manifolds under a \(\mathbb{CP}^2\)-connected sum (Q2184890)

For a smooth \(4\)-manifold \(X\) containing an embedded torus of square \(0\) with a mild condition, \textit{R. Fintushel} and \textit{R. J. Stern} [Invent. Math. 134, No. 2, 363--400 (1998; Zbl 0914.57015)] introduced a knot-surgery technique making a smooth \(4\)-manifold \(X_{K}\) which is homeomorphic but, in general, not diffeomorphic to \(X\) for any non-trivial knot \(K\) in \(S^{3}\). In this paper under review, it is proved that if \(X\) is a smooth \(4\)-manifold containing a codimension zero submanifold (a double node neighborhood of \((S^{1}\times S^{1})\times \{0\}\)) obtained from \((S^{1}\times S^{1})\times D^{2}\) by attaching two \((-1)\)-framed \(2\)-handles along \(S^{1}\times \{\ast\}\times\{\ast\}\) \(\subset \partial ((S^{1}\times S^{1})\times D^{2})\), then all \(X_{K}\) obtained by a knot-surgery along \(S^{1}\times S^{1} \times\{\ast\}\) \(\subset (S^{1}\times S^{1})\times D^{2}\) become mutually diffeomorphic after a connected sum with \(\mathbb{CP}^{2}\) (Theorem 1.1). As a corollary of this result, it is proved that, for the simply connected elliptic surface \(E(n)\) with Euler characteristic \(12n\) and any knot \(K\) in \(S^{3}\), \(E(n)_{K} \#\mathbb{CP}^{2}\) is diffeomorphic to \(\# (2n)\mathbb{CP}^{2}\# (10n-1)\overline{\mathbb{CP}^{2}}\), that is, \(E(n)_{K}\) is almost completely decomposable (Corollary 1.3).