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 $\pi_m$ for $m<N<n$ (this is unlikely to be true though). 2 updated For a fixed$d$, is there a relationship that homotopy groups of smooth$d$-manifolds must satisfy? The$d=1$case is trivial, but I already don't know how to approach$d=2$.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).
For a fixed $d$, is there a relationship that homotopy groups of smooth $d$-manifolds must satisfy?
The $d=1$ case is trivial, but I already don't know how to approach $d=2$.