Topology of positively curved 8-dimensional manifolds with symmetry

Abstract

We show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank ≥ 2 resembles a rank-one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S8, HP2 or CP4. 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 ≥ 2. As an illustration, we apply our results to various families of 8-manifolds. © 2011 by Pacific Journal of Mathematics.