2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58967There 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.12 pages, no figuresAlgebraic TopologyDifferential Geometry55R15; 55R40; 57T35text