Topology of positively curved 8-dimensional manifolds with symmetry

Anand Dessai

Vol. 249 (2011), No. 1, 23–47

DOI: 10.2140/pjm.2011.249.23

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.

Keywords

positive curvature, torus actions, Euler characteristic, classification of 8-manifolds

Mathematical Subject Classification 2000

Primary: 53C20, 57R19, 57S25

Milestones

Received: 18 September 2009

Revised: 10 February 2010

Accepted: 27 February 2010

Published: 3 January 2011