 Solution (source code)

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= Solution

\b[This sequence cannot occur.] <Poincare duality> and the <universal coefficient theorem for cohomology> imply
$$
H_3(M;\mathbb Z)\cong H^1(M;\mathbb Z)\cong\operatorname{Hom}(H_1(M;\mathbb Z),\mathbb Z)=0.
$$
Thus a nonzero rank-two third <homology> group is incompatible with the proposed first <homology> group.

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