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 $\mathbb{R}\mathbf{P}^{13}$. 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.

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Received 16 August 2016

Accepted 1 September 2018

Published 10 May 2021