2 Ted Chinburg ,

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-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 3Sasakian metrics discovered in [BGM]. They are defined on the Eschenburg biquotients Ea,b,c = diag(z , z, zc)\SU(3)/ diag(z, 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: