= Solution
\b[Yes:] real projective three-space $\mathbb{RP}^3$ has these groups. Its usual cell structure has one cell in each dimension zero through three, with boundary maps $\partial_1=0$, $\partial_2=2$ and $\partial_3=0$. Hence its integral <homology> is $\mathbb Z$ in degrees zero and three, $\mathbb Z/2$ in degree one, and zero in degree two. It is a <connected> closed oriented <manifold>: the <antipodal map> on $S^3$ preserves <orientation>, so the sphere's <orientation> descends to the quotient.
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