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Positive curvature and symmetry in small dimensions. (English) Zbl 1446.53024

Extending existing work in small dimensions, A. Dessai [Pac. J. Math. 249, No. 1, 23–47 (2011; Zbl 1228.53046)] computed the Euler characteristic signature, and elliptic genus for 8-manifolds of positive sectional curvature in the presence of torus symmetry. He also computed the diffeomorphism type by restricting the 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 16. In particular, the authors obtain a first characterization of the Cayley plane in such a setting. In the second part, they study a closely related family of manifolds called positively elliptic manifolds, and prove a conjecture of Halperin in this context for dimensions up to 16 or Euler characteristics up to 16.

MSC:

53C20 Global Riemannian geometry, including pinching
57N65 Algebraic topology of manifolds
57R19 Algebraic topology on manifolds and differential topology

Citations:

Zbl 1228.53046
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