Cheeger Manifolds and the Classification of Biquotients

By B. Totaro. 2002

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A closed manifold is called a biquotient if it is diffeomorphic to K\G/H for some compact Lie group G with closed subgroups K and H such that K acts freely on G/H. Every biquotient has a Riemannian metric of nonnegative sectional curvature. In fact, almost all known manifolds of nonnegative curvature are biquotients. The only known closed manifolds of nonnegative sectional curvature which were not defined in this way are those found by Cheeger [6] in 1973, and Grove and Ziller [16] in 2000. In order to understand these constructions better, this paper analyzes which of the Cheeger and Grove-Ziller manifolds are actually diffeomorphic to biquotients. In the process, we develop a general approach to the classification of biquotient manifolds. A priori, it is hard to determine whether a given manifold is a biquotient, because there is no obvious upper bound on the dimension of the groups involved. We give a procedure which allows us to reduce the dimension of the groups needed to describe a given manifold as a biquotient. A surprising application is that Gromoll-Meyer’s example of an exotic 7-sphere which is a biquotient [15] is the only exotic sphere of any dimension which is a biquotient. More generally, we classify all biquotients which are simply connected rational homology spheres of any dimension (Theorem 6.1). The classification of biquotients which are simply connected rational homology spheres has been given independently by Kapovitch and Ziller [17]. More generally, they classify all simply connected biquotients whose rational cohomology ring is generated by one element. Another application of our general classification theory for biquotients is the precise determination of which Cheeger manifolds, the connected sums of two rank-one symmetric spaces with any orientations, are diffeomorphic to biquotients (Theorem 2.1). Some of the Cheeger manifolds were known to be biquotients, such as CP2# − CP2 (the nontrivial S2-bundle over S2), but we find that many of the other Cheeger manifolds are also biquotients, such as CP2#CP2. On the other hand, the Cheeger manifold CP4#HP2 is not diffeomorphic (or even homotopy equivalent) to a biquotient. The positive result that some Cheeger manifolds such as CP2#CP2 are biquotients implies that they have nonnegatively curved Riemannian metrics with various good properties that were not clear from Cheeger’s construction. First, the new metrics are real analytic. Further, the new metrics determine a natural complex-analytic structure on the whole tangent bundle, since Aguilar [2] showed that biquotients have this property; my paper [26] has a weaker result which applies to all the Cheeger manifolds. Finally, the geodesic flow for the new metrics on the Cheeger manifolds has zero topological entropy. Paternain [23] proved the latter property for Cheeger’s metric on CP2#CP2, but now we know it for some metric on a large class of the Cheeger manifolds. Finally, our general classification results imply that there are only finitely many

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fd3b142b4a292eb8f567c2c07eac92ede062a8b9

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In Geometry and Topology

  1. Y. Félix
  2. W. Ziller
  3. V. Kapovitch
  4. B. Totaro
  5. Anton Petrunin
  6. Burkhard Wilking
  7. A. Lytchak
  8. Koichi Nagano

In Mathematical Physics

  1. Y. Félix
  2. W. Ziller
  3. V. Kapovitch
  4. B. Totaro
  5. Koichi Nagano
  6. A. Lytchak

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