Infinitely generated $\pi_2(M)$ where $M$ is closed smooth manifold

Question

I am looking for a closed smooth manifold with infinitely generated $\pi_2$. I know there is an easy example with some universal cover i think, but all i have in my head is $S^2\vee S^1$, which is not even a topological manifold…
Thanks in advance!

Answer 1

Consider the complex projective plane $\mathbb{CP}^2$ and take a connected sum with a $4$-torus, i.e. take $X = \mathbb{CP}^2 \# (S^1)^4$. This is a closed $4$-manifold. Its universal cover $\tilde{X}$ is a connected sum of copies of countably many $\mathbb{CP}^2$ (which is simply connected) so $\pi_2(X) = \pi_2(\tilde{X}) = H_2(\tilde{X})$ is a direct sum of countably many copies of $\mathbb{Z}$ and is therefore not finitely generated.