A new class of homology and cohomology 3-manifolds

TL;DR

The paper constructs specific 3-manifolds that exhibit homology and cohomology properties dependent on chosen prime sets, revealing new distinctions in manifold theory based on coefficient rings.

Contribution

It introduces a novel class of 3-manifolds with prime-dependent homology and cohomology properties, expanding understanding of manifold invariants.

Findings

Existence of manifolds with prime-specific homology properties

Manifolds are not homology with integer or rational coefficients

Demonstrates prime-dependent behavior in 3-manifold invariants

Abstract

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 $q\not\in \mathcal{ P}$. Moreover, $M_{\mathcal{P}}$ is neither a homology nor cohomology 3-manifold with coefficients in $\mathbb{Z}$ or $\mathbb{Q}$.