Fake 13-projective spaces with cohomogeneity one actions (Q2025817)

Fake real projective spaces are manifolds that are homotopy equivalent but not diffeomorphic to standard real projective spaces. First examples thereof were discovered by \textit{M. W. Hirsch} and \textit{J. W. Milnor} [Bull. Am. Math. Soc. 70, 372--377 (1964; Zbl 0201.25601)] as quotients of standard 5- and 6-spheres embedded in Milnor's exotic 7-spheres under certain free involutions. Similarly, \textit{P. Rajan} and \textit{F. Wilhelm} [Bull. Aust. Math. Soc. 94, No. 2, 304--315 (2016; Zbl 1364.53040)] detected some standard 13- and 14-spheres embedded in Shimada's exotic 15-spheres [\textit{N. Shimada}, Nagoya Math. J. 12, 59--69 (1957; Zbl 0145.20303)] with quotients that are homotopy equivalent to \(\mathbb{R}\mathrm{P}^{13}\) and \(\mathbb{R}\mathrm{P}^{14}\), respectively, and they also showed that some of the quotients in the 14-dimensional examples are not diffeomorphic to \(\mathbb{R}\mathrm{P}^{14}\), i.\,e.\ they are fake projective spaces. The first main result of the publication under review is that some of the 13-dimensional homotopy projective spaces \(P^{13}\) found by Rajan and Wilhelm are fake projective spaces as well. Further main results are concerned with invariant metrics of non-negative sectional curvature on fake projective spaces. It is known that the 5-dimensional Hirsch-Milnor fake projective spaces \(P^5\) admit cohomogeneity one actions by \(\mathsf{SO}(2)\times \mathsf{SO}(3)\) [\textit{M. W. Davis}, Am. J. Math. 104, 59--90 (1982; Zbl 0509.57029)] and that all of the \(P^5\)'s carry \(\mathsf{SO}(2)\times \mathsf{SO}(3)\)-invariant metrics of non-negative sectional curvature [\textit{K. Grove} and \textit{W. Ziller}, Ann. Math. (2) 152, No. 1, 331--367 (2000; Zbl 0991.53016), p.334]. Similarly, it is shown here that all Rajan-Wilhelm \(P^{13}\)'s admit cohomogeneity one actions by \(\mathsf{SO}(2)\times \mathsf{G}_2\), but in contrast to the 5-dimensional case the authors now prove the following alternative fact: None of the \(P^{13}\)'s supports an \(\mathsf{SO}(2)\times \mathsf{G}_2\)-invariant metric of non-negative sectional curvature. In fact, it is shown that the 5-dimensional case is rather special in this regard: If a homotopy sphere admits a non-negatively curved metric that is invariant under a cohomogeneity one action, then the sphere is a standard sphere \(\mathbb{S}^n\) and either the action is linear, i.\,e.\ a sub-action of the standard action of \(\mathsf{SO}(n+1)\), or \(n=5\) and the action is nonlinear by \(\mathsf{SO}(2)\times \mathsf{SO}(3)\). Moreover, the authors show that the 13-dimensional Rajan-Wilhelm fake projective spaces are \(\mathsf{SO}(2)\times \mathsf{G}_2\)-equivariantly diffeomorphic to quotients of Brieskorn varieties by involutions. This reformulation is the key technical resource in the proofs of the main results.