What is $\pi_3(S^2\times S^1)$?

What is the third homotopy group of $S^2\times S^1$? Here $S^n$ means the $n$-dimensional sphere.

I am a physicist and did not take courses on too mathematical topics. I just need a fact of this in a recent research project. I thought some mathematicians or students from the mathematical department may know the result. Details on derivation could be skipped (for me).

We may identify: $$\pi_3(S^2\times S^1)\cong \pi_3(S^2)\oplus \pi_3(S^1),$$ via projection onto the two factors.

We know $\pi_3(S^1)\cong \pi_3(\mathbb{R})$ as this is unchanged by passing to the universal cover. As $\mathbb{R}$ is contractible, we get $\pi_3(\mathbb{R})=0$.

For the other factor, $\pi_3(S^2)\cong \pi_3(S^3)$, via the long exact sequence associated to the Hopf fibration, and using the fact that the second and third homotopy groups of the fiber $S^1$ are trivial (for the reason given above):

$$ \begin{array}{cccccc} \to&\pi_3(S^1)\cong 0& \to& \pi_3(S^3)&\to&\pi_3(S^2)\\ \to&\pi_2(S^1)\cong0&\to \end{array} $$

We then have $\pi_3(S^3)\cong H_3(S^3;\mathbb{Z})\cong\mathbb{Z}$, via the Hurewicz homomorphism.

The net result is $\pi_3(S^2\times S^1)\cong\mathbb{Z}$, with the generator being the Hopf map composed with inclusion: $$S^3\stackrel{h}\to S^2 \hookrightarrow S^2\times S^1.$$

In simple terms, the Hopf map is the map from $$S^3=\{(z,w)|\,\,z\bar{z}+w\bar{w}=1\}\subseteq \mathbb{C}^2,$$ to $$S^2=\{[z:w]|\,\,(z,w)\in \mathbb{C}^2 \backslash \{(0,0)\}\},$$ which maps $(z,w)\mapsto [z:w]$.