Fake 13-projective spaces with cohomogeneity one actions

摘要

We show that some embedded standard 13-spheres in Shimada's exotic 15-spheres have 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 RP13. As observed by F. Wilhelm and the second named author in [RW], the Davis SO(2) x 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 Z(2) quotients of certain Brieskorn varieties, and that the Davis SO(2) x 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 SO(2) x SO(3)-cohomogeneity one actions. In contrast, we show that the P(13)s do not support SO(2) x G(2)-invariant metrics with non-negative sectional curvature.