Is S^1 × S^2 homeomorphic to a familiar topological space?

[owlpride]

circle x sphere = ?

Is the product space $$S^1 \times S^2$$ related (e.g. homeomorphic or homotopy equivalent) to a more familiar topological space? I am currently looking at maps from $$S^1 \times S^2$$ into other spaces, and I am having a really hard time visualizing what I am doing. Any thoughts appreciated.

 

[torquil]

Since you can visualize $$S^1$$ and $$S^2$$ by themselves, you should be able to get good impression of the whole space. Or du you need to actually see it?

How about this: Consider $$S^2$$ as the unit sphere $$|x|=1$$ in $$\mathbb{R}^3$$. Then just make it thicker, and identify points on the outer edge with points on the inner edge (along the radius).

Torquil

 

[owlpride]

Well, I am looking at smooth maps $$f: S^1 \times S^2 \rightarrow S^2$$. Then $$f^{-1}(z)$$ is a union of circles, which may or may not be linked. How exactly would I go about visualizing knots in $$S^1 \times S^2$$, and especially their relative position to each other?

 

[torquil]

Well, I am looking at smooth maps $$f: S^1 \times S^2 \rightarrow S^2$$. Then $$f^{-1}(z)$$ is a union of circles, which may or may not be linked. How exactly would I go about visualizing knots in $$S^1 \times S^2$$, and especially their relative position to each other?

Sorry, I don't have anything useful to say about that.

Torquil