A New Class of Homology and Cohomology 3-Manifolds

We show that for any set of primes \({\mathcal{P}}\) there exists a space \({M_{\mathcal{P}}}\) which is a homology and cohomology 3-manifold with coefficients in \({\mathbb{Z}_{p}}\) for \({p \in \mathcal{P}}\) and is not a homology or cohomology 3-manifold with coefficients in \({\mathbb{Z}_q}\) for \({q \not\in \mathcal{P}}\). In addition, \({M_{\mathcal{P}}}\) is not a homology or cohomology 3-manifold with coefficients in \({\mathbb{Z}}\) or \({\mathbb{Q}}\).