Irreducible four-manifolds with small Euler characteristics (Q1913477)

The author investigates two smooth, closed, simply-connected 4-manifolds \(Q\) and \(R\) constructed by \textit{Gompf} [``A new construction of symplectic manifolds'', preprint)] that are topologically \(3\mathbb{C}\mathbb{P}^2\# 16\mathbb{C}\mathbb{P}^2\) and \(3\mathbb{C}\mathbb{P}^2\# 14\mathbb{C}\mathbb{P}^2\). With the standard smooth structure all Donaldson invariants vanish; the author shows however that with the given structure \(Q\) and \(R\) have some non-vanishing SO(3) Donaldson invariants. This is shown by first establishing a Floer-type product formula. He then constructs two families of simply connected closed mutually non-diffeomorphic 4-manifolds \(Q_{2n+1}\) and \(R_{2n+1}\) of the same homeomorphism type of \(Q\) and \(R\) respectively.