Topology of Non-negatively Curved Manifolds

An important question in the study of Riemannian manifolds of positive sectional curvature is how to distinguish manifolds that admit a metric with non-negative sectional curvature from those that admit one of positive curvature. Surprisingly, if the manifolds are compact and simply connected, all known obstructions to positive curvature are already obstructions to non-negative curvature. On the other hand, there are very few known examples of manifolds with positive curvature. They consist, apart from the rank one symmetric spaces, of certain homogeneous spaces G/H in dimensions 6, 7, 12, 13 and 24 due to Berger [Be], Wallach [Wa], and Aloff-Wallach [AW], and of biquotients K\G/H in dimensions 6, 7 and 13 due to Eschenburg [E1],[E2] and Bazaikin [Ba], see [Zi] for a survey. Recently, a new example of a positively curved 7-manifold was found which is homeomorphic but not diffeomorphic to the unit tangent bundle of S, see [GVZ, De]. And in [PW] a method was proposed to construct a metric of positive curvature on the Gromoll-Meyer exotic 7-sphere. Among the known examples of positive curvature there are two infinite families: in dimension 7 one has the homogeneous Aloff-Wallach spaces, and more generally the Eschenburg biquotients, and in dimension 13 the Bazaikin spaces. The topology of these manifolds has been studied extensively, see [KS1, KS2, AMP1, AMP2, Kr1, Kr2, Kr3, Sh, CEZ, FZ]. There exist many 7-dimensional positively curved examples which are homeomorphic to each other but not diffeomorphic, whereas in dimension 13, they are conjectured to be diffeomorphically distinct [FZ]. In contrast to the positive curvature setting, there exist comparatively many examples with non-negative sectional curvature. The bi-invariant metric on a compact Lie group G induces, by O’Neill’s formula, non-negative curvature on any homogeneous space G/H or more generally on any biquotient K\G/H. In [GZ1] a large new family of cohomogeneity one manifolds with non-negative curvature was constructed, giving rise to non-negatively curved metrics on exotic spheres. Hence it is natural to ask whether, among the known examples, it is possible to topologically distinguish manifolds with non-negative curvature from those admitting positive curvature. The purpose of this article is to address this question. There are many examples of non-negatively curved manifolds which are not homotopy equivalent to any of the known positively curved examples simply because they have different cohomology rings. But recently new families of non-negatively curved manifolds were discovered [GZ2] which, as we will see, give rise to several new manifolds having the same cohomology ring as the 7-dimensional Eschenburg spaces. Recall that the Eschenburg biquotients are defined as Ek, l = diag(z k1 , z2 , z3)\ SU(3)/ diag(z1 , z2 , zl3)−1,