Topology of positively curved 8-dimensional manifolds with symmetry

Abstract

In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank $\ge 2$ resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic then M is rationally isomorphic to a rank one symmetric space. For torsion-free manifolds we derive a much stronger classification. We also study the bordism type of 8-dimensional manifolds of positive sectional curvature and symmetry rank $\ge 2$. As an illustration we apply our results to various families of 8-manifolds.

24 pages, a gap in the proof of Theorem 1.1. filled following a suggestion by M. Masuda, some typos corrected, minor changes in the exposition, references added, 18 Dec.09: a few typos corrected

References in corpus (6)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Examples of Riemannian Manifolds with non-negative sectional curvature
• An exotic sphere with positive sectional curvature
• A positively curved manifold homeomorphic to T${}_1$S${}^4$
• On positively curved 4-manifolds with S^1-symmetry