Cartesian product of $S^1$ and $S^2$ Can anyone help me imagine $S^1 \times S^2$? I understand that $T^2=S^1 \times S^1$, but I don't know what to do with the spheres. I am not even sure if it's in 3D. Thank you!

Here is something you could try to visualize to help: When you do $S^1 \times S^1$ to get the torus, one way you could have imagined that was by taking a $S^1$ and gluing it to another $S^1$, and keep gluing more $S^1$ until you get a tube. Then you glue the first $S^1$ to the last $S^1$ to close the ends of the tube, and you get your torus.
Now to do $S^2 \times S^1$. You can hold a $S^2$ in your hand, and wrap it inside another $S^2$. Then you wrap another one over that, and keep wrapping more $S^2$ until you have an Earth-sized ball, only with a hollow core. Finally, you glue the outside of the Earth's surface to the inside of the hollow center.
If this is hard to imagine, you could also go back to the torus construction, but instead of making a "tube" by gluing $S^1$s, grab a pencil and paper and draw a circle, then a slightly bigger circle around that, and keep going until you've drawn a disc with a hole in the middle. This is topologically equivalent to the tube and you make the torus by gluing the hole in the annulus to the outside rim of the annulus. 
Now for the higher dimension, instead of a big 2-ball with a little 2-ball missing from the center, you have a big 3-ball with a little 3-ball missing from the center.
Happy visualizing
