On complexes equivalent to S3-bundles over S4

Abstract

S3-bundles over S4 have played an important role in topology and geometry since Milnor showed that the total spaces of such bundles can be exotic spheres. Until recently, there was only one exotic sphere known to admit a metric of nonnegative sectional curvature. In a recent paper, K. Grove and W. Ziller constructed metrics of nonnegative curvature on the total space of S3-bundles over S4⁠. They also asked for a classification of these bundles up to homotopy equivalence, homeomorphism and diffeomorphism. They also asked the following question: the Berger space, Sp(2)/Sp(1), is a 7-manifold that has the cohomology ring of an S3-bundle over S4⁠, but does it admit the structure of such a bundle? In this paper we show that any 2-connected, 7-manifold with finite fourth cohomology group is PL-homeomorphic to an S3-bundle over S4 if and only if its linking form is equivalent to a standard form. As a corollary we show that the Berger space has the PL type of an S3- bundle over S4⁠. We also derive necessary and sufficient conditions for any CW complex to be homotopy equivalent to an S3-bundle over S4⁠. One of our results has also been obtained independently by D. Crowley and C. Escher.