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# Fake $$13$$-projective spaces with cohomogeneity one actions

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### 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.

### Most cited references9

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### Positively curved cohomogeneity one manifolds and 3-Sasakian geometry

(2008)
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### Cohomogeneity one manifolds with positive Ricci curvature

(2002)
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