Cheeger manifolds and the classification of biquotients
  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. Biquotients are a major source of examples of Riemannian manifolds with nonnegative sectional curvature. We prove several classification results for biquotients:   (1) We classify all simply connected rational homology spheres which are diffeomorphic to biquotients. For example, the Gromoll-Meyer exotic sphere is the only exotic sphere of any dimension which is a biquotient.   (2) We determine exactly which Cheeger manifolds, the connected sums of two rank-one symmetric spaces, are diffeomorphic to biquotients. For example, CP^2 # CP^2 is a biquotient, but CP^4 # HP^2 is not.   (3) There are only finitely many diffeomorphism classes of 2-connected biquotients in each dimension. 
Burt Totaro