On simply connected noncomplex 4-manifolds (Q688613)

We define a sequence \(\{X_ n\}_{n\geq 0}\) of homotopy equivalent smooth simply connected 4-manifolds, not diffeomorphic to a connected sum \(M_ 1 \# M_ 2\) with \(b_ 2^ +(M_ i)>0\), \(i=1,2,\) for \(n>0\), and nondiffeomorphic for \(n\neq m\). Each \(X_ n\) has the homotopy type of \(7\mathbb{C}\mathbb{P}^ 2 \# 37 \overline{\mathbb{C} \mathbb{P}}^ 2\). We deduce that for all but finitely many \(n\) the connected sum of \(X_ n\) with a homotopy sphere is not diffeomorphic to a connected sum of complex surfaces, complex surfaces with reversed orientations and a homotopy sphere.