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Oct 16, 2016 at 12:42 comment added Omar @IgorBelegradek thanks for your answer.
Oct 16, 2016 at 12:36 vote accept Omar
Oct 16, 2016 at 11:49 comment added Igor Belegradek @Omar: let $P_k$ the principal $S^3$ bundle over $S^4$ corresponding to the integer $k\in \pi_4(BS^3)\cong \pi_3(S^3)$. Then $P_k$ is $2$-connected and $H_3(P_k)$ has order $|k|$, see e.g. arxiv.org/abs/math/0007198, proposition 3.3. Take $k=2$.
Oct 16, 2016 at 10:41 comment added Omar Thank you for your answer but I don't see why such a manifold exists?
Oct 16, 2016 at 10:37 history answered John Rognes CC BY-SA 3.0