Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m ≥ 5) of positive sectional curvature on which an (m-1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By [Wi], these results imply a homeomorphism classification for positively curved n-manifolds (n ≥ 8) of almost maximal symmetry rank [InlineMediaObject not available: see fulltext.].