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.

中文翻译：

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伪造$ 13 $射影空间且具有同质性

我们证明，在岛田奇特的$ 15 $-球体中，一些嵌入式标准$ 13 $-球体具有$ \ mathbb {Z} _2 $商空间$ P ^ {13} $ s，它们是假的真实$ 13 $维投影空间，即，它们是同伦等效的，但与标准$ \ mathbb {R} \ mathbf {P} ^ {13} $并没有同态。正如F. Wilhelm和第二任作者在[RW]中所观察到的那样，对Shimada异域$ 15 $球面上的Davis $ \ mathsf {SO}（2）\ times \ mathsf {G} _2 $动作下降为同质性之一我们证明$ P ^ {13} $ s与某些Brieskorn品种的著名$ \ mathbb {Z} _2 $商，以及Davis $ \ mathsf是同构的{P} {2}上的{SO}（2）\ times \ mathsf {G} _2 $操作与这些Brieskorn商上的已知操作等价微分。$ P ^ {13} $ s是Hirsch–Milnor伪造的$ 5 $维投影空间$ P ^ {5} $ s的声调类似物。K. Grove和W. Ziller表明，$ P ^ {5} $ s接受非负曲率的度量，这些度量相对于Davis $ \ mathsf {SO}（2）\ times \ mathsf {SO}（ 3）$-同质性的一种动作。相反，我们显示$ P ^ {13} $ s不支持$ \ mathsf {SO}（2）\ times \ mathsf {G} _2 $-具有非负截面曲率的不变度量。