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

 

[blue_raver22]

Yes, S^1 \times S^2 is related to familiar topological spaces. In fact, it is a well-studied space in topology called the three-dimensional lens space. This space can be thought of as a three-dimensional analog of the torus (S^1 \times S^1) and shares many properties with it. For example, S^1 \times S^2 is a compact, orientable, and simply connected space, just like the torus.

One way to visualize S^1 \times S^2 is as a solid torus with the boundary (S^1 \times S^1) "filled in" with a two-dimensional sphere. This is similar to how we can visualize the torus as a solid circle with the boundary (S^1) "filled in" with a one-dimensional circle.

Furthermore, S^1 \times S^2 is homeomorphic to other familiar spaces such as the three-dimensional projective space (P^3) and the three-dimensional real projective space (RP^3). So, even though it may seem difficult to visualize at first, S^1 \times S^2 is a well-studied and related to other familiar topological spaces. I hope this helps with your understanding and visualization of this space.