Positive curvature and symmetry in small dimensions

Abstract

Extending existing work in small dimensions, Dessai computed the Euler characteristic, signature, and elliptic genus for [Formula: see text]-manifolds of positive sectional curvature in the presence of torus symmetry. He also computes the diffeomorphism type by restricting his results to classes of manifolds known to admit non-negative curvature, such as biquotients. The first part of this paper extends Dessai’s calculations to even dimensions up to [Formula: see text]. In particular, we obtain a first characterization of the Cayley plane in such a setting. The second part studies a closely related family of manifolds called positively elliptic manifolds, and we prove a conjecture of Halperin in this context for dimensions up to [Formula: see text] or Euler characteristics up to [Formula: see text].