Positive curvature and torus symmetry in small dimensions, I -- Dimensions 10, 12, 14, and 16

Abstract

This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds in dimensions up to 16. Many of the calculations are sharp and they require less symmetry than previous classifications. When restricted to certain classes of manifolds that admit non-negative curvature, these results imply diffeomorphism classifications. Also studied is a closely related family of manifolds called positively elliptic manifolds, and we prove the Halperin conjecture in this context for dimensions up to 16 or Euler characteristics up to 16.

48 pages, 8 tables. In Version 2, some proofs are streamlined and other minor improvements and corrections are made

References in corpus (7)

• Examples of Riemannian Manifolds with non-negative sectional curvature
• An exotic sphere with positive sectional curvature
• The classification of compact simply connected biquotients in dimension 6 and 7
• Positive curvature and rational ellipticity
• The classification and curvature of biquotients of the form $Sp(3)//Sp(1{)}^2$
• An exotic sphere with positive curvature
• Homogeneous spin Riemannian manifolds with the simplest Dirac operator

Cited by in corpus (2)

• Torus actions on manifolds with positive intermediate Ricci curvature
• Homology versus homotopy in fibrations and in limits