Why is there no compact manifold without boundary with the following homology groups?

Why is there no compact manifold without boundary with the following
homology groups?

I've been studying homology groups, and this question is stumping me:
Prove there can be no compact manifold $X$ without boundary whose homology
groups are $$H_i(X) = \left\{ \begin{array}{ll} \mathbb{Z} & i = 0 \\
\mathbb{Z}_3 & i=1 \\ 0 & i = 2 \\ \mathbb{Z}_2 & i=3 \\ 0 & i\geq 4
\end{array} \right.$$
I tried creating a chain complex in order to look at differential maps,
and $H_2(X) = 0$ helps gives injectivity to one of the maps, but I'm not
seeing how to prove no such manifold can exist.