Fake \(13\)-projective spaces with cohomogeneity one actions

We show that some embedded standard \(13\)-spheres in Shimada's exotic \(15\)-spheres have \(\mathbb{Z}_2\) quotient spaces, \(P^{13}\)s, that are fake real \(13\)-dimensional projective spaces, i.e., they are homotopy equivalent, but not diffeomorphic to the standard \(\mathbb{R}\mathrm{P}^{13}\). As observed by F. Wilhelm and the second named author in [RW], the Davis \(\mathsf{SO}(2)\times \mathsf{G}_2\) actions on Shimada's exotic \(15\)-spheres descend to the cohomogeneity one actions on the \(P^{13}\)s. We prove that the \(P^{13}\)s are diffeomorphic to well-known \(\mathbb{Z}_2\) quotients of certain Brieskorn varieties, and that the Davis \(\mathsf{SO}(2)\times \mathsf{G}_2\) actions on the \(P^{13}\)s are equivariantly diffeomorphic to well-known actions on these Brieskorn quotients. The \(P^{13}\)s are octonionic analogues of the Hirsch-Milnor fake \(5\)-dimensional projective spaces, \(P^{5}\)s. K. Grove and W. Ziller showed that the \(P^{5}\)s admit metrics of non-negative curvature that are invariant with respect to the Davis \(\mathsf{SO}(2)\times \mathsf{SO}(3)\)-cohomogeneity one actions. In contrast, we show that the \(P^{13}\)s do not support \(\mathsf{SO}(2)\times \mathsf{G}_2\)-invariant metrics with non-negative sectional curvature.