Topology contents to be studied for higher Physics

[sphyrch]

I have Munkres' book on Topology. For higher Physics (beyond standard model, string theory, etc.) I know we need to have an understanding of differential geometry, etc. that assume knowledge in topology.

My question is how much should I study from Munkres' book? I know that it is useful to study till Tychonoff's theorem (chapters 1 to 5), and some of the chapter on algebraic topology.
Do I have to study anything else besides that?

Thanks

 

[homeomorphic]

I have Munkres' book on Topology. For higher Physics (beyond standard model, string theory, etc.) I know we need to have an understanding of differential geometry, etc. that assume knowledge in topology.

You don't necessarily need topology to learn basic differential geometry. But, yes, string theory uses fairly heavy topology. I'm a topologist, not a string theorist, but I know something about what kind of math is used there, even though I don't know much string theory.

My question is how much should I study from Munkres' book? I know that it is useful to study till Tychonoff's theorem (chapters 1 to 5), and some of the chapter on algebraic topology. Do I have to study anything else besides that?

I'm not sure you need all that stuff, even. Chapters 1 and 2 in Munkres are really the core material that you use over and over again in topology--connectedness and compactness. Another important concept is partitions of unity. Other things you can look up as needed. Part 2 is good material to know--the fundamental group and covering spaces. When you start reading a book like Munkres, you might want to try to learn it all just because it's there, but you can move a lot faster if you just concentrate on the most important bits that will be used later on. I think I might have used Tychonoff's theorem once or twice outside of topology, but I use connectedness and compactness on a daily basis. If you want to learn more analysis, you might use different parts of the book than if you were just wanting to go deep into geometry or topology.

String theory draws on more advanced algebraic topology than the basic Munkres Topology book--homology, cohomology, homotopy theory, characteristic classes, fiber bundles, K-theory, and index theory. For that, there are quite a few books. Hatcher's Algebraic Topology is good (most of the material aside from the appendices would be relevant). And Munkres also wrote a separate book about Algebraic topology. I wouldn't worry about the other stuff too much, just yet, but the next thing I would read after learning the contents of Hatcher would be the first chapter of his unfinished book about vector-bundles, available free on his website (as is the algebraic topology book itself).

 

[micromass]

If you want to learn the things for the sake of differential geometry, then perhaps you should look at "Introduction to topological manifolds" by Lee. It's a very good book and it contains about everything you need to know of topology (and then some more). It's a more difficult book than Munkres though, but in my opinion it's a much better book.

 

[sphyrch]

Thanks homeomorphic; that gives me a better perspective on what's required of topology.

@micromass : If possible, could you also please suggest what topics from Lee's book would be sufficient for me to study?

I'm actually in a rush - not that I would at all compromise on proper understanding of the required concepts, but I'm averse to spending time on things that I wouldn't need for higher Physics.

Furthermore, how do you suggest I go about it? Should I just read the standard theorems and do problems, or, in addition to doing the problems, should I try to prove even the given standard theorems after reading definitions?

Thanks again...

 

[micromass]

Certainly do the first 4 chapters, these chapters constitute a basic course in topology, so you should work through them.

And it's best to read the definitions, the theorems AND the proofs. Not doing the proofs will not teach you how to do things in topology.

 

[homeomorphic]

Certainly do the first 4 chapters, these chapters constitute a basic course in topology, so you should work through them.

Not sure if I'd agree, especially for someone who is in a hurry and is more focused on physics. Chapters 1 and 2 will get you pretty far, plus maybe the sections on regular spaces and normal spaces, Urysohn's Lemma, partitions of unity. You can always go back and cover the rest of chapters 3 or 4 later--once you've done chapters 1 and 2, you have what you need to start studying a lot more stuff, like algebraic topology, which will reinforce those concepts.

 

[micromass]

So you're saying that things like compactness, connectedness, quotient spaces, subspaces are all not important?? That is what topology is all about!

 

[homeomorphic]

Oh, I checked my copy of Munkres. I was off by one chapter. I thought I remembered the chapters. I mean chapters 1-3.

So, what I meant to say is chapter 4 could be mostly skipped.