For a fixed $d$, is there a relationship that between the homotopy groups of smooth $d$-manifolds must satisfy?d$-manifolds?
The $d=1$ case is trivial, but I already don't know how to approach $d=3$ (I should have said that the case of $d=2$ is simple as well, since there is only a sphere to consider, but I don't know how to formulate the property of "having the same homotopy groups as $S^2$" in a simpler way).
Note about the discussion on the comments: it's unreasonable to expect an easy complete characterization of homotopy groups of $S^2$, even less for other manifolds. But I think one could try some partial relations. An interesting relationship would be: for some $d$, the groups
$\pi_n$ can be determined from groups
$m<N<n$ (this is unlikely to be true though).