A problem of Kreck on Poincaré manifolds (Q1434520)

\textit{M. Kreck} [Arch. Math. 77, No. 1, 98--106 (2001; Zbl 0990.57008)] called for an investigation of the converse of the Poincaré conjecture by defining a simply connected closed manifold \(M\) to be a Poincaré manifold if whenever \(N\) is another simply connected closed manifold with the same homology as that of \(M\), \(N\) must be homeomorphic to \(M\). He asked for which \(p\) and \(q\) in the metastable range \(3\leq p < q\leq 2p-3\) it holds that \(S^p\times S^q\) is a Poincaré manifold. The authors answer this by giving the following necessary and sufficient conditions: (1) \(p=3, 5, 6, 7 \mod 8\), (2) for \(p+1 < q\), \(\pi_{q-1}(SO(p+1)) = 0\), and (3) for \(p+1 = q\), the inclusion induced homomorphism \(\pi_p(SO(p))\to \pi_p(SO(p+1))\) is trivial. The techniques come from differential topology. By using calculations of \textit{C. Hoo} and \textit{M. Mahowald} [Bull. Am. Math. Soc. 71, 661--667 (1965; Zbl 0142.40801)], the authors provide concrete examples (e.g. \(S^5\times S^6\) is a Poincaré manifold). Related results are obtained in the piecewise linear category by using block bundle theory.