We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence X i of Alexandrov spaces with curv k Gromov-Hausdorff converging to a compact Alexandrov space X , X i is homeomorphic to X for all large i.

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Abstract. We classify all biquotients whose rational cohomology rings are generated by one element. As a consequence we show that the Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient.

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We show that almost nonnegatively curved m-manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C.m/-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.