Topological properties of Eschenburg spaces and 3-Sasakian manifolds

TL;DR

This paper investigates the topological and geometric properties of Eschenburg spaces and 3-Sasakian manifolds, revealing new examples of homeomorphic but not diffeomorphic spaces and manifolds with multiple 3-Sasakian metrics.

Contribution

It constructs new examples of Eschenburg and 3-Sasakian spaces with unique topological and geometric features, including multiple 3-Sasakian metrics on the same manifold.

Findings

Existence of many homeomorphic but not diffeomorphic 3-Sasakian spaces.

First example of a manifold with two non-isometric 3-Sasakian metrics.

Construction of many pairs of positively curved Eschenburg spaces that are homeomorphic but not diffeomorphic.

Abstract

The authors examine topological properties of the 7-dimensional Eschenburg biquotients diag(z^k1,z^k2,z^k3)\SU(3)/diag(z^l1,z^l2,z^l3). A subfamily of these spaces carry a 3-Sasakian metric. The authors show that among this subfamily there exist many 3-Sasakian spaces which are homeomorphic but not diffeomorphic. In addition, they construct a pair of 3-Sasakian spaces which are diffeomorphic, thus giving the first example of a manifold which carries two non-isometric 3-Sasakian metrics. This answers an open question of C. Boyer and K.Galicki. Among the general family, the authors construct many pairs of positively curved Eschenburg spaces which are homeomorphic but not diffeomorphic. Such pairs were first constructed by Kreck-Stolz among the special subfamily of Aloff-Wallach spaces.