Positive curvature and rational ellipticity

Abstract

Simply connected manifolds of positive sectional curvature are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, ie to have only finitely many non-zero rational homotopy groups. In this article, we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include an upper bound on the Euler characteristic and new evidence for a couple of well-known conjectures due to Hopf and Halperin. We also prove a conjecture of Wilhelm for even-dimensional manifolds whose rational type is one of the known examples of positive curvature.