Topological properties of Eschenburg spaces and 3-Sasakian manifolds

We examine topological properties of the seven-dimensional positively curved Eschenburg biquotients and find many examples which are homeomorphic but not diffeomorphic. A special subfamily of these manifolds also carries a 3-Sasakian metric. Among these we construct a pair of 3-Sasakian spaces which are diffeomorphic to each other, thus giving rise to the first example of a manifold which carries two non-isometric 3-Sasakian metrics. Mathematics Subject Classification (2000) 53C25 · 57R19 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. Christine Escher was supported by a grant from the Association for Women in Mathematics. Wolfgang Ziller was supported by the Francis J. Carey Term Chair, and Ted Chinburg and Wolfgang Ziller were supported by a grant from the National Science Foundation. T. Chinburg (B) · W. Ziller University of Pennsylvania, Philadelphia, PA 19104, USA e-mail: [email protected] W. Ziller e-mail: [email protected] C. Escher Oregon State University, Corvallis, OR 97331, USA e-mail: [email protected]