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Homeomorphism Classification of positively curved manifolds with almost maximal symmetry rank

Fuquan Fang·Xiaochun Rong
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Abstract

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>4) of positive sectional curvature on which a (m-1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By a result of Wilking, these results imply a homeomorphism classification for positively curved n-manifolds (n>7) of almost maximal symmetry rank [\frac{n-1}2].

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Fuquan Fang, Xiaochun Rong. 2003-11-12. Homeomorphism Classification of positively curved manifolds with almost maximal symmetry rank. https://arxiv.org/abs/math/0311183

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