An orientable 4-manifold with completely nontrivial 2-type

Question

I'm looking for a closed, orientable 4-manifold $M$ with completely nontrivial 2-type. By "completely nontrivial", I mean the following:

• The fundamental group $\pi_1(M)$ has to be nontrivial.
• The second homotopy group $\pi_2(M)$ has to be nontrivial.
• The action of $\pi_1(M)$ on $\pi_2(M)$ has to be nontrivial.

A nonorientable example would be $\mathbb{RP}^2 \times S^2$, thanks to Chris Schommer-Pries.

Answer 1

Take any simply-connected oriented $4$-manifold $M$ and perform surgery on an embedded $S^0$ (i.e. take out two disks and glue together the remaining boundaries).

The universal cover of this can be explicitly described as $\mathbb{Z}$ copies of $M$ with those disks removed, glued together in sequence, with deck transformations acting by translation on summands.

So $\pi_1$ is $\mathbb{Z}$. $\pi_2$ is the same as $H_2$ of the universal cover, which is easily seen to be a sum of $\mathbb{Z}$ many copies of $H_2(M)$. $\pi_1$ acts by translation on summands.

So this construction gives an example, starting with any simply-connected $M$ with nontrivial $H_2$.