For \(n\geq 2\) let \(M\) be an \((2n-2)\)-connected \((4n-1)\)-dimensional Poincaré complex with \(H^{2n}(M;\mathbb{Z})\cong\bigoplus^{\ell}_{k=1}\mathbb{Z}/p_k^{r_k}\mathbb{Z}\) where each \(p_k\) is an odd prime and each \(r_k\geq1\). Then the \(2n\)-skeleton of \(M\) is homotopy equivalent to a wedge of Moore spaces \(M_{2n}\simeq\bigvee^{\ell}_{k=1}P^{2n}(p^{r_k}_k)\). Let \(m\) be the least common multiple of \(\{p_1,\ldots,p_{\ell}\}\). Then \(m=\bar{p}^{\bar{r}_1}_1\cdots\bar{p}^{\bar{r}_s}_s\), where \(\{\bar{p}_1,\ldots,\bar{p}_s\}\) is the set of distinct primes in \(\{p_1,\ldots,p_{\ell}\}\) and \(\bar{r}_j\) is the maximum power of \(\bar{p}_j\) appearing in the list \(\{p_1^{r_1},\ldots,p_{\ell}^{r_{\ell}}\}\). There are homotopy equivalences \(P^{2n}(m)\simeq\bigvee^s_{k=1}P^{2n}(\bar{p}^{\bar{r}_k}_k)\) and \(M_{2n}\simeq P^{2n}(m)\vee\Sigma A\), where \(\Sigma A\) is the wedge of the remaining Moore spaces in \(M_{2n}\). Let \(V\) be the mapping cone of the inclusion \(\Sigma A\hookrightarrow M\). Then \(V\) is a Poincaré Duality complex with \(H^{2n}(V;\mathbb{Z})\cong\mathbb{Z}/m\mathbb{Z}\).
The main results (Theorem 1.1) of this paper are to prove the homotopy equivalences \[ \Omega M\simeq\Omega V\times\Omega((\Sigma\Omega V\wedge A)\vee\Sigma A) \] and \[ \Omega V\simeq\left(\prod^s_{j=1}S^{2n-1}\{\bar{p}_j^{\bar{r}_j}\}\right)\times\Omega S^{4n-1}, \] where \(S^i\{p\}\) is the homotopy fibre of the degree map \(p\colon S^{i}\to S^i\). The first equivalence decomposes \(\Omega M\) into a product of spaces, while the factors can be identified with products of recognizable spaces by the second equivalence. In particular, the second factor \(\Omega((\Sigma\Omega V\wedge A)\vee\Sigma A)\) is homotopy equivalent to the loop space of a wedge of spheres and odd primary Moore spaces.
In Sections 3 and 4 the authors prove the second equivalence by constructing a map \(e\colon\left(\prod^s_{j=1}S^{2n-1}\{\bar{p}_j^{\bar{r}_j}\}\right)\times\Omega S^{4n-1}\to\Omega V\) and showing that it induces a homology isomorphism rationally and with mod-\(p\) localized coefficients for each prime \(p\). Then \(e\) is an integral homology isomorphism and hence a homotopy equivalence due to the Whitehead Theorem. Using this result the authors show that there is a fibration sequence \((\Sigma\Omega V\wedge A)\vee\Sigma A\to M\to V\) which splits after looping, and consequently produces a map \(\Omega V\times\Omega((\Sigma\Omega V\wedge A)\vee\Sigma A)\to\Omega M\) giving the first equivalence. In addition, the authors prove that \[ \Omega(M_{2n}\vee S^{4n-1})\simeq\Omega M\times\Omega((P^{4n-1}(m)\wedge\Omega M)\vee P^{4n-1}(m)) \] in Theorem 1.2, implying that \(\Omega M\) retracts off the loop space of a wedge of Moore spaces and a sphere.
In Sections 5 and 6 the authors give two partial extensions for their results when some of the primes \(p_k\) are 2 (Theorems 5.7 and Proposition 6.1).