Does there exist a smooth compact manifold whose homotopy groups are not finitely generated

Question

Does there exist a smooth compact manifold whose homotopy groups are not finitely generated?

I found a counter-example for topological manifolds, but I did not understand whether it is possible to introduce a smooth structure on it

Answer 1

If $M_1$ and $M_2$ are closed $3$-manifolds with infinite fundamental group, and if $M = M_1 \# M_2$ is the connected sum of $M_1$ and $M_2$, then the group $\pi_2(M)$ is not finitely generated. The proof uses the fact that $\pi_2(M)$ is isomorphic to $\pi_2(\widetilde M)$, where $\widetilde M$ is the universal covering space of $M$, together with the Hurewicz theorem to deduce $\pi_2(\widetilde M) \approx H_2(\widetilde M)$, together with a get-your-hands-dirty calculation to prove that $H_2(\widetilde M)$ is not finitely generated.

The hypothesis on fundamental groups of $M_1$ and $M_2$ can be considerably weakened, really the only situations to avoid are when one of them is $S^3$ and when both of them are $\mathbb RP^3$.

And, by the way, $M$ has a smooth structure, as does any closed 3-manifold.