Fake $13$-projective spaces with cohomogeneity one actions

TL;DR

This paper constructs and analyzes 13-dimensional fake projective spaces as quotients of exotic spheres, showing their geometric properties and symmetries, and contrasting their curvature characteristics with lower-dimensional analogues.

Contribution

It demonstrates that certain 13-dimensional homotopy equivalent spaces are diffeomorphic to Brieskorn quotients and explores their symmetry actions and curvature properties.

Findings

The $P^{13}$ spaces are diffeomorphic to Brieskorn quotients.

They admit cohomogeneity one actions similar to known models.

They do not support non-negative curvature metrics invariant under their symmetry group.

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.