Smoothly embedded 2-spheres and exotic 4-manifolds (Q1329052)

In [Topology 30, 479-511 (1991; Zbl 0732.57010)] \textit{R. E. Gompf} showed that the nuclei \(N_ k\) and \(N_ k(0)\) of the minimal elliptic surface \(V_ k\) and the manifold \(V_ k (0)\) obtained from it by performing a degree 0 logarithmic transformation are homeomorphic but non- diffeomorphic. These 4-manifolds have 2-component link pictures which generalize easily to families \(M(k,l,m)\). Here \(N_ k(0) = M(1,k,1)\) and \(N_ k = M(k,1,1)\). The 3-manifolds \(\partial M (k,l,m)\) are independent of the order of \(k,l,m\). The intersection form of \(M (k,l,m)\) is \(\{\begin{smallmatrix} 0 & 1 \\ 1 & - k \end{smallmatrix}\}\). This paper generalizes Gompf's results as follows: Theorem 1.1. Let \(k,l,m\), \(k \neq l\), be positive integers with either \(k \leq 2\) or \(l \leq 2\), and \(s\) a nonnegative integer. If either (i) \(s \neq 0\) or (ii) \(s = 0\) and \(k \equiv l\) mod 2, then the manifolds \(M(k,l,m) \#s (\overline {\mathbb{C} \mathbb{P}^ 2})\) and \(M(l,k,m) \# s (\overline {\mathbb{C} \mathbb{P}^ 2})\) are homeomorphic but their interiors are not diffeomorphic. The homeomorphism statement follows easily from Freedman's work. For non- diffeomorphism, there are two cases \(k = 1\) and \(k = 2\) to prove, and they are handled by similar arguments. The key idea in each case is that in \(M(k,l,m)\) there is a naturally embedded 2-sphere of square \(-k\). If there were a diffeomorphism of interiors, it would lead to a class \(\alpha\) of square --1 or --2 in a blown up elliptic surface and a corresponding reflection self diffeomorphism \(R_ \alpha\) having properties which contradict results of \textit{R. Friedman} and \textit{J. W. Morgan} [J. Differ. Geom. 27, No. 2, 297-369 (1988; Zbl 0669.57016), No. 3, 371-398 (1988; Zbl 0669.57017)]. Using similar methods the author gives an alternate proof of a result of \textit{S. Akbulut} [An exotic 4- manifold, ibid. 33, 357-361 (1991)] giving two framed knots \(K_ 1, K_ 2\) so that the 4-manifolds \(W_ 1, W_ 2\) obtained by performing surgery on them are homeomorphic but their interiors are not diffeomorphic.