Irreducible exotic ${\mathbb{CP}}^2\#7{\overline{\mathbb{C}}\mathbb{P}}^2$'s with free involutions

Abstract

We construct infinitely many pairwise non-diffeomorphic irreducible smooth four-manifolds homeomorphic to ${\mathbb{CP}}^2\#7{\overline{\mathbb{C}}\mathbb{P}}^2$, each admitting a free, orientation-preserving involution. Their quotients give infinitely many irreducible definite four-manifolds with fundamental group $\mathbb{Z}/2\mathbb{Z}$ and $b_2=3$, filling the corresponding gap in the geography of such examples. The construction of infinitely many distinct examples relies on a novel equivariant torus surgery.