On Complexes Equivalent to $\mathbb{S}^3$-bundles over $\mathbb{S}^4$

Details On Complexes Equivalent to $\mathbb{S}^3$-bundles over $\mathbb{S}^4$ On Complexes Equivalent to $\mathbb{S}^3$-bundles over $\mathbb{S}^4$

2000-04-03

arxiv.org/abs/math/0004013v1

Mathematics

Algebraic Topology

12 pages, no figures

Scientific paper

There has been renewed interest in $\mathbb{S}^3$-bundles over $\mathbb{S}^4$ since K. Grove and W. Ziller constructed metrics on nonnegative curvature on the total spaces of these bundles. In this paper we write down necessary and sufficient conditions for a CW complex to be homotopy equivalent to such a bundle. We also show that for a manifold homotopy equivalent to such a bundle, in certain cases, there is no obstruction to homeomorphism. We use this to show that the Berger manifold, $\text{Sp}(2)/\text{Sp}(1)$, is PL-homeomorphic to such a bundle. This question was raised in the paper by Grove and Ziller since this manifold, which admits a normal homogeneous metric of positive sectional curvature, has the cohomology ring of such a bundle. The principal technique used is the study of the Serre spectral sequence of various fibrations.

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