Reference on Geometric Topology

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

There is a good course by Jacob Lurie in here:

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