Where can I find information on S^3?

[RicciFlow]

Hello everyone,

I am looking for some resources that could help me understand interesting properties of S^3 (or S^n really). I am doing a small project for my topology course and in my studies I have found that I lack knowledge about S^3. Are there any good texts or online lecture notes somewhere that cover this topic? In particular I might find certain quotient spaces of S^3 useful. Thus far I have been out of luck both online and at my school library.
Thank you all in advance for any suggestions!

-RF

 

[fzero]

The Hopf fibration of $S^3$ illustrates a lot of important geometry, while also being relevant to certain quotient spaces, like $S^3/S^1 = S^2$ or the Lens spaces $S^3/(\mathbb{Z}/p)$. That wiki seems to be very detailed, but Nakahara's Geometry, Topology and Physics should also have a review, along with all of the background that is relevant.

 

[quasar987]

On R^4 there is a multiplicative group structure that make R^4 into a skew field Q called the quaternions.

Sitting naturally inside R^4 as the unit sphere, one can check that S^3 is a subgroup for this multiplicative group structure on R^4.

Therefor, S^3 has a natural group structure. This group structure is furthermore compatible with its topology in the sense that the group operations of multiplication and taking inverse are continuous. We say S^3 is a topological group. In fact, S^3 is, aside from S^1, the only sphere admitting a topological group structure!

In looking for group acting on S^3, you may thus look for multiplicative subgroups of S^3, or of Q itself and intersect them with S^3.

 

[RicciFlow]

The Hopf fibration of $S^3$ illustrates a lot of important geometry, while also being relevant to certain quotient spaces, like $S^3/S^1 = S^2$ or the Lens spaces $S^3/(\mathbb{Z}/p)$. That wiki seems to be very detailed, but Nakahara's Geometry, Topology and Physics should also have a review, along with all of the background that is relevant.

Excellent! This is precisely the kind of thing I was hoping for! The Hopf Fibration looks very interesting and I believe I will spend some time learning about it now. Also, thanks for the book recommendation, I will check it out soon. Thanks again!

On R^4 there is a multiplicative group structure that make R^4 into a skew field Q called the quaternions.

Sitting naturally inside R^4 as the unit sphere, one can check that S^3 is a subgroup for this multiplicative group structure on R^4.

Therefor, S^3 has a natural group structure. This group structure is furthermore compatible with its topology in the sense that the group operations of multiplication and taking inverse are continuous. We say S^3 is a topological group. In fact, S^3 is, aside from S^1, the only sphere admitting a topological group structure!

In looking for group acting on S^3, you may thus look for multiplicative subgroups of S^3, or of Q itself and intersect them with S^3.

Oh! Dang! I feel as though I should have known this since I know of the definition of S^3 in terms of quaternions. Thank you for pointing it out though! I have a feeling this too will be very useful for me. Thanks!

 

[lavinia]

look at the book, "Three Dimensional Geometry and Topology" Voume 1 by William Thurston