The Hantsche obstruction to embedding a 3-manifold $M$ in a homology 4-sphere is a $\mathbb Q/\mathbb Z$-valued bilinear form on the torsion subgroup of $H_1(M;\mathbb Z)$. If you were to use (co)homology with rational coefficients this would be invisible to you.
The Hantsche obstruction to embedding a 3-manifold $M$ in a homology 4-sphere is a $\mathbb Q/\mathbb Z$-valued bilinear form on the torsion subgroup of $H_1(M;\mathbb Z)$. If you were to use (co)homology with rational coefficients this would be invisible to you.