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Riemannian manifolds with positive sectional curvature have been a frequent topic of global Riemannian geometry for over 40 years. Nevertheless, there are relatively few known examples of such manifolds. The purpose of this article is to study the topological properties of some of these examples, the so-called Eschenburg spaces, in detail.In addition to positively curved metrics, some Eschenburg spaces also carry another special geometric structure, namely a 3-Sasakian metric, i.e. a metric whose Euclidean cone is Hyperkähler [BG]. 3-Sasakian spaces are interesting since they are Einstein manifolds and are connected to several other geometries: They admit an almost free, isometric action by SU(2) whose quotient is a quaternionic Kähler orbifold. The twistor space of this orbifold, which can be viewed as an S 1 -quotient of the 3-Sasakian manifold, carries a natural Kähler-Einstein orbifold metric with positive scalar curvature.3-Sasakian structures are rare and rigid, in fact the moduli space of such metrics on a fixed manifold consists of at most isolated points. This motivated C. Boyer and K. Galicki to pose the question in [BG][Question 9.9, p. 52] whether a manifold can admit more than one 3-Sasakian structure. Natural candidates for such examples are the 3-Sasakian metrics discovered in [BGM]. They are defined on the Eschenburg biquotients E a,b,c = diag(z a , z b , z c )\SU(3)/ diag(z a+b+c , 1, 1), where a, b, c are positive, pairwise relatively prime integers. The simplest topological invariant of these spaces is the order of the fourth cohomology group, which is a finite cyclic group of order r = ab + ac + bc. By studying further topological invariants of these manifolds we show:Theorem A. For r ≤ 10 7 , there is a unique pair of 3-Sasakian Eschenburg spaces E a,b,c which are diffeomorphic to each other, but not isometric. It is given by (a, b, c) = (2279, 1603, 384) and (2528, 939, 799) with r = 5143925.The two 3-Sasakian metrics are non-isometric since the isometric action by SU(2) has cyclic isotropy groups of order a + b, a + c and b + c. In [BG] they also asked whether two 3-Sasakian manifolds can be homeomorphic to each other but not diffeomorphic. This happens frequently among the 3-Sasakian Eschenburg spaces. There are 96 such pairs for r ≤ 10 7 , the first one of which is given by (a, b, c) = (171, 164, 1) and (223, 60, 53) for r = 28379. See Table 4.5 for the next 4 such pairs.The manifolds E a,b,c also carry a metric of positive sectional curvature, although the 3-Sasakian metric never has positive curvature. They are special cases of the more general Date: December 27, 2017.

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 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

We give a classification of a specific family of seven-dimensional manifolds, the generalized Witten manifolds up to homeomorphism and diffeomorphism. Using an approach suggested by J. Shaneson, we develop a modified surgery theory which fully classifies these manifolds. In contrast to previous approaches, this surgery theory is designed to be more readily applicable to higher dimensions. The family contains examples of manifolds which admit Einstein metrics and are homeomorphic but not diffeomorphic. These particular manifolds occur naturally in differential geometry and are of great interest to both differential geometers and physicists. Mathematics Subject Classification (2000). 55R15, 55R40, 57T35.

Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.