List all $i$ for which there is a closed orientable $6$-manifold $ M$ with $H_i(M) = \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$.
So, obviously this has to do with Poincare Duality. We can't have $i=0$ or $i=6$ by definition of a manifold. So we are left with the cases $i=1$ through $5$, which reduce to $i=1, 2,$ and $3$, since the rank of $H_1$ is equal to the rank of $H_5$ by Poincare Duality (and thus for $H_2$ and $H_4$, too). I am thinking that setting up the exact sequences here gives me some kind of equation which will lead to a contradiction in some of those cases, but I'm not able to see why yet. Can I get a push in the right direction, and an evaluation of my thoughts so far?
