As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain either a point or an open dense set of points at which all 2-planes have positive curvature. We study infinite families of biquotients defined by Eschenburg and Bazaikin from this viewpoint, together with torus quotients of $S^3 \x S^3$.Comment: V3 An error has been discovered in Section 3. This section has been removed and the Introduction modified accordingly; V4 published versio

Abstract. In this work, it is shown that a simply-connected, rationallyelliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved.

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of nonnegatively curved manifolds which contain points at which all 2-planes have positive curvature. We show that there are generalisations of the well-known Eschenburg spaces and quotients of S 7 S 7 which admit metrics with this property. 53C20; 57S25, 57T15It is an unfortunate fact that for a simply connected manifold which admits a metric of nonnegative curvature there are no known obstructions to admitting positive curvature. Petersen and Wilhelm [21] propose that the Gromoll-Meyer exotic 7-sphere admits positive curvature, which would be the first exotic sphere known to exhibit this property.In this paper we are interested in the study of manifolds which lie "between" those with nonnegative and those with positive sectional curvature. It is hoped that the study of such manifolds will yield a better understanding of the differences between these two classes. Recall that a Riemannian manifold .M; h ; i/ is said to have quasipositive curvature (resp. almost positive curvature) if .M; h ; i/ has nonnegative sectional curvature and there is a point (resp. an open dense set of points) at which all 2-planes have positive sectional curvature.Theorem A (i) There exists a free circle action and a free S 3 action on S 7 S 7 such that each of the respective quotients M 13 WD S 1 n.S 7 S 7 / and N 11 WD S 3 n.S 7 S 7 / admits a metric with almost positive curvature. (ii) If M 13 and N 11 are equipped with the metrics from (i), then there exist Riemannian submersions M 13 ! CP 3 and N 11 ! S 4 such that in each case the fibre is S 7 and the bundle is nontrivial but shares the same cohomology ring as the corresponding product.We use the Pontrjagin class to distinguish M 13 and N 11 from the respective products in Theorem A(ii). Moreover, in each case the induced metric on the base is positively curved.It has been conjectured by F Wilhelm that, if M is a positively curved Riemannian manifold, the dimension of the fibre of a Riemannian submersion M ! B must be less than the dimension of the base. Theorem A shows that this is false when the hypothesis is weakened to almost positive curvature.Other than the Gromoll-Meyer exotic 7-sphere (see While Wilking [30] has shown that it is not possible in general to deform quasipositive curvature to positive curvature, it is still unknown whether this can be achieved in the simply connected case or whether one can always deform quasipositive curvature to almost positive curvature.

In a recent article, the authors constructed a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature. Each member of this family is the total space of a Seifert fibration with generic fibre S 3 and, in particular, has the cohomology ring of an S 3 -bundle over S 4 . In the present article, the linking form of these manifolds is computed and used to demonstrate that the family contains infinitely many manifolds which are not even homotopy equivalent to an S 3 -bundle over S 4 , the first time that any such spaces have been shown to admit non-negative sectional curvature.

Let M n , n ∈ {4, 5, 6}, be a compact, simply connected n-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on M n by a torus T n−2 is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, nonnegatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.The classification of (compact) Riemannian manifolds (M n , g) with positive or nonnegative (sectional) curvature is a notoriously difficult problem. One of the few successes in this quest occurs when one considers such manifolds equipped with an effective action by a suitably large group G of isometries. The ambiguity of the term "suitably large" allows various classification results to be achieved (see, for example, [BB, GS1, GS2, GWZ, Ve, Wa, Wi1, Wi2] and the surveys [Gr,Wi3]).There are, in fact, two parts to the classification program. First is the topological classification, the goal of which is to determine, up to diffeomorphism, all possible positively or non-negatively curved manifolds on which G can act. The second part is the equivariant classification, where the goal is to determine all possible actions of G on a given positively or non-negatively curved manifold up to equivariant diffeomorphism.This approach to the classification problem is inspired by the work of Hsiang and Kleiner [HK]. They showed that a simply connected, four-dimensional manifold (M 4 , g) with positive curvature admitting an effective, isometric circle action must be homeomorphic to either S 4 or to CP 2 . If (M 4 , g) is assumed to have only non-negative curvature then, by Kleiner [Kl] and Searle and Yang [SY], M 4 must be homeomorphic to one of S 4 , CP 2 , CP 2 # ± CP 2 or S 2 × S 2 . In both situations, homeomorphism is improved to diffeomorphism by appealing to work of Fintushel [Fi1, Fi2], Pao [Pao] and Perelman's proof of the Poincaré conjecture [Pe1, Pe2].The existence of an effective, isometric circle action is equivalent to the rank of the isometry group Iso(M n , g) being positive. The success in dimension four suggests that it may be beneficial to consider the topological classification when "largeness" of our group G of isometries refers to the symmetry rank of (M n , g), defined as the rank of Iso(M n , g) and denoted symrank(M n , g). Indeed, Grove and Searle [GS1] showed that if (M n , g) is positively curved, then symrank(M n , g) ⌊ n+1 2 ⌋. Moreover, if the symmetry