@Aleksandar Milivojević, I've got your point. The post is edited.

@abx, according to Wall's classification theorem(Classification problem in Topology V), $M$ is diffeomorphic to $S^3 \times S^3$ if $H_2(M; \mathbb Z) =0$ and $H_3(M; \mathbb Z)$ is torsion-free (These conditions for cohomologies are missing in the original post -- now edited). Hence $M$ is a Calabi–Eckmann manifold but its canonical class is not trivial. And For the part "$b_3 >0$" with Serre duality, I was mistaken. Now edited.

@Aleksandar Milivojević: Could you give a reference for your statement "Topologically any .... homology sphere"?

Great answer! What if simply-connectedness is added to the conditions? Maybe it's asking too much?

There are smooth Fano threefolds with $H_V^3=12$ with index one. Do you mean that the K3 surface in $P(1,1,1,3)$ cannot be embedded into any of those Fano threefolds?

You're right. Thanks for the answer.