Recommendation for Low-dimensional topology textbook

Question

Can anyone here recommend a low-dimensional topology textbook that contains knot theory and 3,4-manifolds?Or should I look for these subjects in separate textbooks?

Answer 1

My book preferences are different from the other answer! My preferences are the following:

Knot theory: Lickorish "An introduction to knot theory". You don't need much background apart from singular homology/fundamental groups.

3-manifolds: Saveliev "Lectures on the topology of 3-manifolds". Some of the earlier content overlaps with Lickorish, but it's also good! Again if you know some stuff about singular homology you'll be able to read this book. It's quite selective about content though - if you're interested in things like contact topology, geometrisation etc, you'll need other sources.

4-manifolds: Scorpan "Wild world of 4-manifolds". The previous answer also mentioned this book - I like it a lot! The earlier chapters develop theory within the algebraic topology framework, but eventually we need more tools: later on Seiberg-Witten gauge theory is introduced and a bunch of wild 4-dimensional results are proven (e.g. uncountably many smooth structures on R^4)

Answer 2

These are diverse subjects studied with different tools. You will need a strong background in algebraic topology, as well as manifold topology (PL or differentiable, most likely the latter), to do well with the material.

Here are some very good introductory texts in each.

Knot theory: Rolfsen's "Knots and links". Background: in addition to a good full course in algebraic topology, know differentiable manifolds through the isotopy extension theorem.

3-manifold topology: Hempel's book is the classic. Hatcher's short set of notes is a good substitute, though it doesn't cover as much. At some point you should read Peter Scott's paper on geometries of 3-manifolds.

The theory of 4-manifolds is too diverse to be well-discussed in one book. One should read Gompf and Stipsicz, "4-manifolds and Kirby Calculus". After that you can try to feel around for your particular taste. Some people like the book by Scorpan but I haven't read it.

Answer 3

In addition to what others have mentioned, I would like to add Hatcher's "Notes on Basic 3-Manifold Topology" which don't cover too much material but are in my opinion the best introduction to 3-Manifolds around.