Fake $13$-projective spaces with cohomogeneity one actions
نویسندگان
چکیده
We show that some embedded standard $13$-spheres in Shimada's exotic $15$-spheres have $\mathbb{Z}_2$ quotient spaces, $P^{13}$s, are fake real $13$-dimensional projective i.e., they homotopy equivalent, but not diffeomorphic to the $\mathbb{R}\mathrm{P}^{13}$. As observed by F. Wilhelm and second named author [RW], Davis $\mathsf{SO}(2)\times \mathsf{G}_2$ actions on descend cohomogeneity one $P^{13}$s. prove $P^{13}$s well-known quotients of certain Brieskorn varieties, equivariantly these quotients. The octonionic analogues Hirsch-Milnor $5$-dimensional $P^{5}$s. K. Grove W. Ziller showed $P^{5}$s admit metrics non-negative curvature invariant with respect \mathsf{SO}(3)$-cohomogeneity actions. In contrast, we do support \mathsf{G}_2$-invariant sectional curvature.
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ژورنال
عنوان ژورنال: Communications in Analysis and Geometry
سال: 2021
ISSN: ['1019-8385', '1944-9992']
DOI: https://doi.org/10.4310/cag.2021.v29.n3.a5