Cell structure of $S^2 \times S^1$

Question

Can anyone please provide the cell structure of $S^2 \times S^1$? I know that there are one cell in each dimension from 0 to 3 but I am not sure about the attaching maps.

Thanks in advance.

Answer 1

$S^3=S^1\wedge S^2=S^1\times S^2/S^1\vee S^2$. In other words, $S^1\times S^2 = S^1\vee S^2 \cup_\varphi e^3$, where $\varphi$ is the attaching map $$\partial e^3=(\partial e^1\times e^2)\cup(e^1\times\partial e^2)\to S^1\vee S^2,$$ which sends $\partial e^2$ and $\partial e^1$ to the base point of $S^1\vee S^2$.