Abstract
Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.
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The first author was supported by a grant of the German Research Foundation. The second author was partially supported by National Science Foundation Grant DMS-1045292.
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Amann, M., Kennard, L. Topological properties of positively curved manifolds with symmetry. Geom. Funct. Anal. 24, 1377–1405 (2014). https://doi.org/10.1007/s00039-014-0300-9
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DOI: https://doi.org/10.1007/s00039-014-0300-9
Keywords and phrases
- Positive curvature
- Symmetry
- Euler characteristic
- Betti numbers
- Hopf conjecture
- Elliptic genus
- Symmetric spaces