Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 16 / 3 / c / Solution

This sequence cannot occur. The same duality and coefficient calculation gives $H_3(M)=0$ when $H_1(M)=0$. More generally the third homology of a closed oriented four-manifold is torsion-free, whereas the proposed nonzero third group is torsion. Either observation rules it out.