On complexes equivalent to \(\mathbb{S}^3\)-bundles over \(\mathbb{S}^4\) (Q2725068)

Let \(X\) be a simply connected CW-complex with suitable cohomology properties specified in the paper [the reviewer observes that the authors do not include, e.g., the requirement that \(H^i(X;\mathbb Z)=0\) if \(i>7\)]. The main theorem states that a necessary and sufficient condition for \(X\) to be homotopy equivalent to an \(S^3\)-bundle over \(S^4\) (here \(S^k\) means the \(k\)-sphere) is that two certain conditions are fulfilled: one of them is expressed in terms of a secondary cohomology operation, and the other in terms of a linking form for \(X\). Several related results on simply connected smooth \(7\)-dimensional manifolds are also presented. NEWLINENEWLINENEWLINE\textit{K. Grove} and \textit{W. Ziller} [Curvature and symmetry of Milnor spheres, Ann. Math. (2) 152, No. 1, 331-367 (2000; Zbl 0991.53016)] mention that the paper under review partially answers a question posed by them, by showing that \`\` the positively curved Berger space \(B^7 = SO(5)/SO(3)\) is PL-homeomorphic to an \(S^3\) bundle over \(S^4\)''.