Article

An exotic sphere with positive sectional curvature

06/2008;
Source: arXiv

ABSTRACT

We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.

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    • "By constructing invariant metrics of non-negative sectional curvature on cohomogeneity one manifolds with codimension two singular orbits, and applying this to the associated principle bundles of the Milnor spheres, Grove and Ziller [14] proved that all 10 Milnor spheres admit metrics with non-negative sectional curvature. In 2008, Petersen and Wilhelm [35] showed that there is a metric on the Gromoll–Meyer sphere with positive sectional curvature. "

    Full-text · Dataset · Jun 2015
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    • "Further, until the recent example due to Grove, Verdiani, and Ziller [17] and Dearicott [7], all known examples of compact manifolds with positive sectional curvature were diffeomorphic to biquotients. See [5] [1] [37] [11] [12] [4] [31]. Furthermore, all known examples of manifolds with almost or quasipositive curvature are diffeomorphic to biquotients. "
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    ABSTRACT: We classify all compact simply connected biquotients of dimension 4 and 5. In particular, all pairs of groups $(G,H)$ and embeddings $H\rightarrow G\times G$ giving rise to a particular biquotient are classified.
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    • "They are what is called rationally elliptic spaces, meaning that their total rational homotopy π * (·)⊗Q is finite dimensional. (See [43] for a survey of examples, see Dearricott [8], Grove–Verdiani–Ziller [17], and Petersen–Wilhelm [28] for two new examples in dimension seven and see [10] for an compendium on rational homotopy theory.) Rational ellipticity on its own has interesting consequences. "
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    ABSTRACT: Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm.
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