Are S^1 x S^2 & Familiar Topological Spaces Related?

[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