Fake $13$-projective spaces with cohomogeneity one actions

Abstract

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.

39 pages, one appendix

References in corpus (1)

• Almost Nonnegative Curvature on Some Fake $R{P}^{6}s$ and $R{P}^{14}s$