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Geometric topology is more motivated by objects it wants to prove theorems about. Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional phenomena being special is due to the existence of a big tool called the Whitney Trick, which allows one to readily convert certain problems in manifold theory into (sometimes quite complicated) algebraic problems. The thing is the Whitney trick fails in dimensions 4 and lower.

As to my background, I've learnt Boothby's book "An Introduction to Differential Manifolds ...". I recently want to dive in some depth into Geometric Topology. But I found the literature is quite a mess. Could anyone suggest a textbook or at least a sequence of books and papers that leads to the frontier of this field?

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Also posted on MO: mathoverflow.net/questions/93408 Please don't post on two fora at the same time and if you do so then at least provide links in order to avoid duplication of efforts. –  t.b. Apr 7 '12 at 11:57

4 Answers 4

up vote 12 down vote accepted

Stillwell's book Classical Topology and Combinatorial Group Theory is a good first place to start to get a feel for the techniques of geometric topology. If you want to get your feet wet in the world of $4$-manifolds, there's a great book called The Wild World of $4$-manifolds by Scorpan which could serve as a source of further papers for you to look at. For $3$ dimensions, I would start to learn some knot theory. There are many good books on this.

In general, you will still need to know algebraic topology, even if you are only interested in the geometric side of things. In my opinion, Hatcher's book Algebraic Topology does a superb job explaining the subject while maintaining geometric intuition.

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Great answer! This will keep me busy for some time. –  caozhu Apr 7 '12 at 12:56

This answer is meant to complement Jim's.

Guillemin and Pollack's Differential Topology text is a great start that's not too specialized any any particular direction. Once you've got some basic algebraic topology background, you can start to link up a lot of basic notions via Guillemin and Pollack (Poincare duality, intersection theory).

A lot of geometric topology is motivated or informed by constructions from the general theory of manifolds. Milnor's Morse Theory is an excellent read once you're done Guillemin and Pollack. Learning a bit of the basics of Lie Groups would be good at this stage. From there you're ready for things like the strong Whitney embedding theorem, and the h-cobordism theorem. Kosinski's Differential Manifolds book and Milnor's Lectures on the h-cobordism theorem are a very good pair of books to read at that time.

To get your feet wet in knot theory I'm a big fan of Rolfsen's Knots and links. It's a great book for self-learning, as there's oodles of computations left for the reader. Hatcher's 3-manifolds notes will get you started with some basic 3-manifold theory. Thurston's book 3-dimensional geometry and topology followed by Geometry and topology of 3-manifolds will get you started on 3-manifold theory. Bonahon's new book Low-dimensional geometry is similar to Thurston's book but is perhaps a little gentler to the reader. For 4-manifolds, Kirby's The topology of 4-manifolds is a good start. Gompf and Stipsicz 4-manifolds and kirby calculus gets you going from there.

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Good suggestions, all. –  Grumpy Parsnip Apr 8 '12 at 19:32

There is a good course by Jacob Lurie in here:

http://www.math.harvard.edu/~lurie/937.html

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Part of the problem with geometric topology is that there still is no wide agreement as to what exactly the field consists of. Some aspects are parts of differential topology/geometry,some belong to algebraic geometry and others belong properly to algebraic topology. Frankly,I think the term "geometric topology" refers to an approach rather then an actual separate field. To me, a good working definition of geometric topology is that it refers to those aspects of topological spaces-particularly manifolds-that can be studied either by combinatorial methods,as in classical low dimensional topology, or by the direct generalization of these methods,such as piecewise linear structures and their characterization by their associated cell decompositions.
The best beginning source for a geometric approach to topology is the wonderful book Classical Topology And Combinatorial Group Theory by John Stillwell. In addition to the very modern course notes by Jacob Lurie recommended by user32240 above,I can heartily recommend the following links page, which contains a wealth of lecture notes and online texts on topology-including not only links to 2 courses taught by Lurie, but many other classic jewels in the field including hard to find lecture notes on differential topology dating to the 1960's by C.T.Wall, the classic text on PL topology by Hudson, complete sets of all the classical lecture notes by John Milnor at Princeton (!) ,Dennis Sullivan's classic 1973 MIT lectures and much much more. I think you'll find it immensely helpful.

http://www.maths.ed.ac.uk/~aar/surgery/notes.htm

Lastly, I'd like to add a recent discovery: John Francis at Northwestern University has posted a number of lecture notes on geometric and advanced algebraic topology at his personal page-they look quite good and are worth a look:

http://www.math.northwestern.edu/~jnkf/

Clarification: Many of the sources above are quite advanced,but many-sucb as the notes by Zeeman and Wall-are not and should be accessible to beginners. The book by Hudson is the logical next step in the subject after Stillwell and a course in differential topology.The notes by Francis are fairly advanced and require at least a semester of a graduate course in algebraic topology a la Hatcher for their full comprehension,but there's a wealth of material on the applications of the Whitney trick to low-dimensional manifolds there,which the OP specifically asked about. That should get you started-good luck!

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I have a policy of not responding to you, but I'll go against my better judgement and do so (but I won't respond further). I am one of the currently two downvoters. I did so for two reasons. First, your "definition" of geometric topology is really bad. Geometric topology has nothing to do with "combinatorial methods", and most current work on it (eg gauge theory, or Thurston-style 3-manifold topology, or surgery theory) is distinctly non-combinatorial. If you don't have actual knowledge of a subject, please refrain from trying to shovel the bs around. Second, the sources you (continued) –  Adam Smith Jun 16 '12 at 17:57
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recommend are wildly inappropriate given the OP's stated background. I can't imagine that you have actually tried to read eg Sullivan's MIT notes if you think they are appropriate for someone who just read Boothby. –  Adam Smith Jun 16 '12 at 18:00
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@Mathemagician1234, you seem to be leaving your mark on a lot of posts that I'm looking at. As someone who is currently going through some GT [and listening to the posters above], I was actually curious as to how many of these resources you've gone through yourself and how you find them. It seems [and I am NOT implying that you did this this] that many of your resources were either recommended by a previous poster, or on the first page of a google search on the subject --- this kind of thing is arguably not as helpful as those who have been through the work. On the other hand, –  james Jun 22 '12 at 23:54
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(cont.) if you have read them, I'd be interested in your opinions on each. –  james Jun 22 '12 at 23:54
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@Mathemagician1234 : If you had really read most of the books you recommend, then you would be capable of answering actual technical questions about advanced mathematics. As it is, you have no technical answers here with positive vote totals, and the technical answers you have given are uniformly incoherent/wrong and concern beginning undergraduate topics. You surely must understand why this leads people to be skeptical of your knowledge... –  Adam Smith Jun 27 '12 at 22:41

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