Let \(n\geq 2\) and let \(M\) be an \((n-1)\)-connected \((2n+1)\) dimensional closed manifold. Then one can easily see that \[ H_i(M;\mathbb{Z})= \begin{cases} \mathbb{Z} & \text{if }i=0,\ 2n+1 \\ \mathbb{Z}^r\oplus G & \text{if }i=n \\ \mathbb{Z}^r & \text{if }i=n+1 \\ 0 & \text{otherwise} \end{cases} \] where \(G\) is a finite abelian group and \(r\geq 0\) is an integer. In this paper the author studies the homotopy types of the above manifolds \(M\). In particular, he determines the \(p\)-local homotopy groups of \(M\) when a prime \(p\) is not divided by the order of \(G\) and he determines the homotopy type of the loop space \(\Omega M\) explicitly when \(r\geq 1\). More precisely, he shows that if \(r\geq 1\) there is a homotopy equivalence \[ \Omega M\simeq \Omega S^n\times \Omega S^{n+1}\times \Omega (Z\vee (Z\wedge \Omega (S^n\times S^{n+1}))), \] where \(M(G,n)\) is the Moore space of type \((G,n)\) and \(Z\simeq \vee_{r-1}(S^n\vee S^{n+1})\vee M(G,n)\). His proof is based on the careful analysis of quadratic associative algebras and Koszul duality arising from the cohomology ring of \(M\) and he obtains the above results by using the Diamond Lemma and the Poincaré-Birkhoff-Witt Theorem.
For the entire collection see [Zbl 1411.55001].