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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.