Exploring the 3-Sphere: A Beginner's Guide to Higher Dimensions

[Sunfire]

Hello,

I am a novice to manifolds and higher dimensions. Could you recommend a good introductory book(s) or web sites (besides Wikipedia, of which I am aware ) on the subject of the 3-sphere. My goal is to build intuition about the 3-sphere by starting with simple analogies, examples etc.

I am not too familiar with group theory, would like to approach the 3-sphere without group notation, if possible.

Thank you!

 

[homeomorphic]

I'm not sure I understand the fascination with the 3-sphere in particular. The 3-sphere is more like a very specific example, rather than something you'd specifically set a goal to learn all about. That being said, it is a very good example to study in differential geometry and topology.

Sounds to me maybe you should start with a book about curves and surfaces like Elementary Differential Geometry by O'Neill.

It's true that you can view S^3 as the unit quaterions (or an incarnation of SU(2)), but I don't think groups are some kind of huge thing that is used whenever you talk about S^3 that you have to watch out for. That's only in certain contexts. You could talk about its homology/cohomology or homotopy groups, too, of course. There are a lot of different aspects of S^3 and it's not clear what kind of understanding you're after.

 

[Sunfire]

That being said, it is a very good example to study in differential geometry and topology.
... There are a lot of different aspects of S^3 and it's not clear what kind of understanding you're after.

I would like to learn what is the current understanding of the geometry of the 3-sphere and what are the common elements of the 2-sphere and the 3-sphere - e.g. equator; meridians; volume; surface, poles etc. How can one visualize any section of the 3-sphere; is there a software that does various stereographic projections of the 3-sphere/its elements.

If anything else comes to mind, please post here :)

I found this site and liked very much its introduction. Its math is clear and very logical. But then on the subsequent pages it gets very hard to follow. "Inversion"? Why is it done? etc...

 

[homeomorphic]

Its math is clear and very logical. But then on the subsequent pages it gets very hard to follow. "Inversion"? Why is it done? etc...

You'd just have to read more about topology and geometry, I think. I'm not sure what books to recommend aside from O'Neill.

 

[Sunfire]

Thank you for the suggested read. Just checked, our library has a similar book available to download as pdf: Elementary differential geometry / Andrew Pressley. Just went through the whole book with Ctl-F. He doesn't deal with the 3-sphere... Oneill's book shows as available online but the link points to Pressley's book. They do have a hard copy though. Does O'neill have a chapter specifically on the 3-sphere? I couldn't tell by looking at the very condensed TOC they have online. Another one I looked at is Elementary differential geometry / Christian Bär. This one has a 9-page chapter Spherical and hyperbolic geometry, followed by Cartography. Perhaps it is about 2-spheres only...

I will keep looking for similar titles like the one you suggested. It just seems very hard to find something dedicated to the 3-sphere. This is why decided to post the question here.

 

[Sunfire]

Have you heard of The shape of space / Jeffrey R. Weeks; it has a chapter on "hypersphere". Just wondering if there is something better out there.

I so liked the introductory pages of this link, but as mentioned above, I wish the writer wouldn't sharply change his presentation after chapter 2.

 

[homeomorphic]

Well, I just don't think you should be so focused on that one example. My recommendation is to think about learning differential geometry, rather than specifically about the 3-sphere. But it's good to use as an example. You can calculate the curvature of the 2-sphere in like 5 different ways, following a book on curves and surfaces. And you can try to do the same for the 3-sphere if you get to higher-dimensional curvature.

As I said, the 3-sphere is mostly just an example, rather than something people study for its own sake.

 

[WWGD]

Sunfire: maybe you can read up on the Hopf fibration , in which you see the 3-sphere fibered by 1-spheres; this is a locally-trivial bundle of the three-sphere S^3 over the 2-sphere S^2 with fiber S^1.

 

[homeomorphic]

Sunfire: maybe you can read up on the Hopf fibration , in which you see the 3-sphere fibered by 1-spheres; this is a locally-trivial bundle of the three-sphere S^3 over the 2-sphere S^2 with fiber S^1.

Assuming he knows what fibrations and locally-trivial bundles are. I don't think that's quite novice-level.

I'm thinking he might have to look at something like Munkres topology first to learn about homeomorphisms and product spaces and all that.

Here's a book that has some good motivation for vector bundles and fiber bundles:

http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html

The Hopf fibration is just one more piece of evidence that you're not going to understand the 3-sphere ONLY by thinking about the 3-sphere.

 

[ChrisVer]

What is special about a 3 sphere in comparison to a 2sphere?
There is nothing much to change... you embed a 2sphere into the R^{3} euclidean space in order to describe it... for the 3 sphere you analyse everything in the same way, you only have to embed it in the R^{4} euclidean space

 

[Sunfire]

Sunfire: maybe you can read up on the Hopf fibration.

Assuming he knows what fibrations and locally-trivial bundles are. I don't think that's quite novice-level.

Yes, I would love to, provided it doesn't involve group theory beyond the definitions

I understand the benefit of learning math concepts first and then assemble all accumulated knowledge about the 3-sphere in one neat mental package. My original question is - in your personal experience, what have been good resources where you read about the 3-sphere. If you point me to them, I will be able to see for myself how much I am able to process and if additional knowledge is required, I might post other questions at this forum

Thanks for your suggestions so far. If you have sources in mind on the 3-sphere (websites, book chapters, theses, white papers) please post them here. Much appreciate.

 

[WWGD]

Sunfire, you can read on the Hopf fibration; even if it is at this point difficult for you, your interest in the topic will allow you to continue working on it. And bring us your questions.

 

[Sunfire]

Sunfire, you can read on the Hopf fibration; even if it is at this point difficult for you, your interest in the topic will allow you to continue working on it. And bring us your questions.

Thank you for suggesting the Hopf fibration. Like with any other topic, there will be books which are really good to start with... Would you be able to suggest a good book. (I have a fair background in vector/calculus, linear algebra and differential equations.)

 

[homeomorphic]

Yes, I would love to, provided it doesn't involve group theory beyond the definitions

I'm not sure why you are so terrified of group theory if you are up for learning other math. A lot of abstract algebra books might not have the best motivation, but you may be able to find one that's more friendly.

Visual Group Theory by Nathan Carter is an interesting-looking one that I've never read, but I will read it some day if I ever have the time for it (I'm no longer a practicing mathematician, so more marketable knowledge is now my priority).

 

[WWGD]

Thank you for suggesting the Hopf fibration. Like with any other topic, there will be books which are really good to start with... Would you be able to suggest a good book. (I have a fair background in vector/calculus, linear algebra and differential equations.)

I think the Wikipedia article: http://en.wikipedia.org/wiki/Hopf_fibration is not bad for a start. Let me see if I can find other resources.