Topology Resources

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Arariel
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Topology Resources

Postby Arariel » Fri Dec 10, 2010 3:32 am UTC

Does anyone know a good resource (book, lecture series, etc.) for topology? Preferably with a cost of less than $50.

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Re: Topology Resources

Postby Yakk » Fri Dec 10, 2010 1:30 pm UTC

Algebraic, Point-set, introductory, what?
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Re: Topology Resources

Postby MostlyHarmless » Fri Dec 10, 2010 8:54 pm UTC

Topology, by James Munkres. Apparently it's much more expensive than I remembered, but I'm sure you can find a used copy much cheaper than that.

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Re: Topology Resources

Postby Cleverbeans » Fri Dec 10, 2010 9:00 pm UTC

I found Toplogy: Point-Set and Geometric by Paul L. Shick to be an excellent introductory text for self-study. The scope and presentation of the content is significantly different than Munkres book, so either may be appropriate based on your interests and background.
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Re: Topology Resources

Postby Arariel » Sat Dec 11, 2010 1:16 am UTC

All right, thanks.

Yakk: Probably introductory.

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Re: Topology Resources

Postby Yakk » Sat Dec 11, 2010 11:36 am UTC

There is always Rudin:
http://www.amazon.com/Principles-Mathem ... pd_sim_b_2
which doesn't coddle you and covers (point-set) topology from the direction of analysis if I remember correctly. (note: I am not kidding about coddling, and book is mainly an analysis book)

Given the lack of answer, I'm guessing you want point-set. :)

Is there a purpose beyond 'topology is cool'? (and it is cool, but there might be other reasons!)
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Arariel
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Re: Topology Resources

Postby Arariel » Sat Dec 11, 2010 2:05 pm UTC

As long as I can understand it without any prior knowledge of topology, it should be all right.

As for why, I'm trying to cover a few of the math topics with applications in physics.

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Re: Topology Resources

Postby Yakk » Sun Dec 12, 2010 6:15 am UTC

Ah. So you want to go from topology to basic analysis to differential topology, take a gander at lie groups, maybe not category theory, some algebraic topology to understand how "topological sameness" is detected... Information theory (shannon entropy etc) will also be useful if you want to have a foundation for thermodynamics from the discrete side.

The foundations of the above will require rings, groups, fields, linear algebra (reasonably theoretical), basic analysis (to keep yourself sane when you are doing the more advanced stuff)

Does that seem reasonable? Some of the above is beyond me (I am still wrestling with Lie group theory), but it all looks connected to physics from my perspective.
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Re: Topology Resources

Postby doogly » Sun Dec 12, 2010 4:31 pm UTC

Armstrong's Basic Topology is a great place to start. The neat little Dover books, Counterexamples in Topology and Counterexamples in Analysis are also really delightful.
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Re: Topology Resources

Postby Arariel » Mon Dec 13, 2010 3:12 am UTC

Yakk wrote:The foundations of the above will require rings, groups, fields, linear algebra (reasonably theoretical), basic analysis (to keep yourself sane when you are doing the more advanced stuff)

Linear algebra, I got, sorta. Group theory? If that's what you mean, I know a bit. What do you mean with rings and fields?

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Re: Topology Resources

Postby doogly » Mon Dec 13, 2010 3:33 am UTC

Generally algebra books treat all three of those things, with a 1 semester course paying most attention to groups. I'd be surprised though if a book you used for that didn't also have some discussion of rings and fields a bit later.
Rings and Fields add more structure to a set in addition to the group operation. Rings have a second operation, but not every element has an inverse under the 2nd operation. Fields do.
To get going in algebraic topology, you are actually pretty fine only having studied groups. The most important things in algebraic topology are the homotopy, homology and cohomology groups. You could get through the intro in Armstrong and probably through most (all?) of Hatcher's Algebraic Topology without much more. Hatcher is also fabulous, I should mention, though you'd want to do a basic thing first.
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Re: Topology Resources

Postby Yakk » Mon Dec 13, 2010 1:07 pm UTC

You need a bit of ring & field theory for linear algebra and to get your chops set up for understanding algebraic structures (like sigma algebras etc). But really, ring/field/group intro courses tend to be bundled together. Groups are the simplest, structure-wise, but seem to be the one with the richest theory that connects to physics.

Group theory is needed for Lie groups and symmetry groups and the like, which is apparently pretty huge in theoretical physics.

Ie, the monster moonshine conjecture, which connects the largest simple group (with ~ 8E54 elements) to string theory and elliptic curves, was proven in 1998.

http://www.daviddarling.info/encycloped ... cture.html

When you see string theory talk about objects like E8, they are talking about group theory (I think!)

For lie groups, look at this google: http://google.com/search?q=lie+groups+and+physics

Which also contains this link: http://superstringtheory.com/math/math2.html a guide to what kinds of math someone who wants to understand/research theoretical physics needs.
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Re: Topology Resources

Postby doogly » Mon Dec 13, 2010 1:47 pm UTC

Oh yes, all that stuff is exceedingly useful for further work in physics.
Especially if you don't want to be "that guy" who uses math without understanding it. Go ask your average physicist what a Hilbert space is actually defined as, and watch the squirms.
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Re: Topology Resources

Postby Marbas » Mon Dec 13, 2010 9:24 pm UTC

Ah. So you want to go from topology to basic analysis


There are actually books with this basic progression in mind I believe. For example, I have Topology for Modern Analysis from Dover, and it has proved pretty useful. I know there's a more well regarded title in this vein, but currently it's escaping me.
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Re: Topology Resources

Postby theorigamist » Tue Dec 14, 2010 12:04 am UTC

Perhaps you should look at Calculus on Manifolds by Spivak, or Introduction to Manifolds by Loring Tu.

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Re: Topology Resources

Postby Arariel » Tue Dec 14, 2010 12:31 am UTC

So anything I read will probably cover rings and fields, is that right?

EDIT: Turns out I didn't know what a group was, but I found a site that explained groups, rings, and fields, so never mind, then.

Well, thanks for the suggestions. This'll keep me busy over break, I guess. :D

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Re: Topology Resources

Postby Mapar » Tue Dec 14, 2010 2:59 pm UTC

Would you mind posting a link, Arariel? I happen to be interested in the same stuff. :)
I like to tackle some of the more advanced stuff in my spare time, since I might be going into math and/or theoretical physics.
Hi.

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Re: Topology Resources

Postby bobdylan » Tue Dec 14, 2010 5:51 pm UTC

If you didn't know what a group was, you're probably not ready for Algebraic Topology by Hatcher. It is, however, cheap and/or available free online. It's quickly becoming the standard graduate textbook

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Re: Topology Resources

Postby Arariel » Tue Dec 14, 2010 11:51 pm UTC

Mapar wrote:Would you mind posting a link, Arariel? I happen to be interested in the same stuff. :)
I like to tackle some of the more advanced stuff in my spare time, since I might be going into math and/or theoretical physics.


http://www.quadibloc.com/math/abaint.htm

bobdylan wrote:If you didn't know what a group was, you're probably not ready for Algebraic Topology by Hatcher. It is, however, cheap and/or available free online. It's quickly becoming the standard graduate textbook


Well, I know what they are now. Do you have a link?

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Re: Topology Resources

Postby doogly » Tue Dec 14, 2010 11:53 pm UTC

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Re: Topology Resources

Postby Yakk » Wed Dec 15, 2010 12:04 am UTC

Rings and Fields is both about getting acquainted with them, and working on algebraic proof chops.

Knowing what ideals are, understanding what "modding out" does, etc is much easier if you've done it and proven how it works in a comparatively simple framework.

I also suspect "not knowing what a group is" is being used as a proxy. :)
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Arariel
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Re: Topology Resources

Postby Arariel » Wed Dec 15, 2010 12:09 am UTC

Yakk wrote:Rings and Fields is both about getting acquainted with them, and working on algebraic proof chops.

Knowing what ideals are, understanding what "modding out" does, etc is much easier if you've done it and proven how it works in a comparatively simple framework.

I also suspect "not knowing what a group is" is being used as a proxy. :)


In that case, is there anything I can check out for practice problems?

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Re: Topology Resources

Postby doogly » Wed Dec 15, 2010 12:26 am UTC

There are a bunch of great text books that can help you get a deep understanding of groups, rather than a cursory familiarity. I like Artin's "Algebra" a lot. If you don't have access to a library there are some decent free things you can google too.
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Re: Topology Resources

Postby theorigamist » Wed Dec 15, 2010 1:21 am UTC

If you really want to learn algebra, then get Dummit and Foote's Abstract Algebra. It's a staple of any mathematician's library.

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Re: Topology Resources

Postby doogly » Wed Dec 15, 2010 1:22 am UTC

Eh, I didn't like it as much as Artin, and I certainly wouldn't start with it.
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Re: Topology Resources

Postby skullturf » Wed Dec 15, 2010 1:23 am UTC

If we're talking about introductions to abstract algebra for undergraduates, my vote is for Gallian.

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Re: Topology Resources

Postby B.Good » Wed Dec 15, 2010 2:07 am UTC

skullturf wrote:If we're talking about introductions to abstract algebra for undergraduates, my vote is for Gallian.

I've been working through Hungerford's undergraduate text (Abstract Algebra: An Introduction) and I like what I've seen more than what I've done in Gallian's text (I used to work through Gallian). The problems are arranged in terms in difficulty, i.e. the A. Problems are easiest, B. Problems are challenging and C. problems are generally quite difficult, unfortunately, you are not given that luxury with Gallian's text. The subjects covered are very "algebraic", symmetry groups and other geometric concepts are not covered very much at all although they are covered. I also think that the material is covered better in Hungerford's text. You can also get a used copy of Hungerford's text for much less money than a used copy of Gallian's text. That said, Gallian's text is by no means a terrible book, and I'm only comparing Hungerford to Gallian because those are the only two abstract algebra texts that I have used.

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Re: Topology Resources

Postby skullturf » Wed Dec 15, 2010 3:01 am UTC

Gallian might not be the most encyclopedic algebra text, but it's very user-friendly and a joy to learn from.

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Re: Topology Resources

Postby B.Good » Wed Dec 15, 2010 3:38 am UTC

skullturf wrote:Gallian might not be the most encyclopedic algebra text, but it's very user-friendly and a joy to learn from.

I will say that I had fun learning from Gallian's text but I felt that I learned more from Hungerford's and since I was very into math when I started reading Hungerford, I had just as much fun. Since this is more of a topic about Topology, I'll stop my input for now.

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Re: Topology Resources

Postby theorigamist » Wed Dec 15, 2010 5:40 am UTC

doogly wrote:Eh, I didn't like it as much as Artin, and I certainly wouldn't start with it.

Well, as I've said before, you're wrong about not liking Dummit and Foote. I can't compare it to Artin, though, as I've never read Artin. But it's one of the big three algebra texts (along with Lang, and Dummit and Foote), so it's probably good. Better to be safe than sorry: buy both.

As for Gallian and Hungerford, I just don't see the need for a specifically undergraduate algebra textbook when Dummit and Foote exists. Dummit and Foote:
1. goes at a very comfortable pace,
2. has lots of fully-worked-through examples (making it very suitable for self-study),
3. has tons of problems at the end of each section which are very accessible and clearly relate to the section, and
4. covers such a breadth of topics that it will remain useful throughout college, graduate school, and for research purposes.

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Re: Topology Resources

Postby bobdylan » Wed Dec 15, 2010 10:06 am UTC

I really enjoyed Fraleigh's "A First Course in Abstract Algebra", which goes all the way from the definition of a group to field extensions and Galois theory.

The old edition I had and sadly lost also had a short section on simplicial homology, which is what first got me interested in algebraic topology (I'm now writing my thesis). Unfortunately newer editions seem to have dropped it.


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