Communications in Analysis and Geometry

Volume 29 (2021)

Number 3

Fake $13$-projective spaces with cohomogeneity one actions

Pages: 707 – 760

DOI: https://dx.doi.org/10.4310/CAG.2021.v29.n3.a5

Authors

Chenxu He (Department of Mathematics, University of California, Riverside, Calif., U.S.A.)

Priyanka Rajan (Department of Mathematics, University of Notre Dame, South Bend, Indiana, U.S.A.)

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