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Background

I have studied:

  • Algebraic Topology: Hatcher's Algebraic Topology (minus the Additional Topics there). However, I'd definitely like to learn more if the need arises.
  • Differential Topology: Introduction to Smooth Manifolds by John M. Lee; Differential Topology by Guillemin & Pollack; Morse Theory by Milnor. I have recently begun reading Differential Manifolds by Kosinski.
  • I also know much Riemannian, complex and symplectic geometry, if that helps.

What I'm Looking For

There are many interesting fundamental questions about the topology of manifolds:

  • How do we go about classifying topological or differential manifolds of a given dimension?
  • How many differential structures are there on a given topological manifold? In particular, why is the differential structure unique in dimensions $\leq3$?
  • Which manifolds can be triangulated? In particular, why is it that all manifolds are triangulable in dimensions $\leq3$?

I'm looking for a textbook that addresses the above (and similar) questions, if such a book exists. I am aware of the book Foundational Essays on Topological Manifolds, Smoothings, and Triangulations by Kirby and Siebenmann, but it is certainly way too advanced for a beginner like me.

Specific Questions

  1. Would a book entirely devoted to, say, $3$-manifolds, be too specific for what I want to know?
  2. Which topics (among, e.g., surgery theory, PL topology, knot theory, $4$-manifolds) are relevant here?
  3. Is there any order in which I should study these topics? For example, should I study knots before $3$-manifolds and vice versa? Should I study $3$-manifolds before $4$-manifolds?
  4. Could you please suggest an (ordered) list of books to read?
  5. I've heard of some good books. For example, is Thurston's $3$-manifold book too advanced for me?

Please let me know if my question is still too vague. Thanks for any advice!

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  • $\begingroup$ Good and hard goal and good start. Have you solved all the problems in these books? (Lee, Hatcher, Milnor) $\endgroup$
    – C.F.G
    Mar 8 '20 at 18:23
  • $\begingroup$ @C.F.G Thanks! Hatcher's exercises are often too difficult for me. I actually only read it when cramming for an exam. But I'm definitely going to reread it sometime. As far as I recall, there were no exercises in Milnor's Morse Theory. I did do a lot of exercises in Riemannian Geometry, though (say, 70% in Petersen's book). :) $\endgroup$
    – Colescu
    Mar 9 '20 at 2:20
  • $\begingroup$ OMG. It is my dream that solving 70% of exercises because I am so lazy. Are you really undergraduate? good luck and be care about virus. $\endgroup$
    – C.F.G
    Mar 9 '20 at 6:12
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    $\begingroup$ @C.F.G Yeah, I spent more than a whole month doing that book. And it's very nice of you to say that! The epidemic in our country is now under control, and I hope the same happens to yours soon. :) $\endgroup$
    – Colescu
    Mar 9 '20 at 9:37
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Regarding

  • How do we go about classifying topological or differential manifolds of a given dimension?
  • How many differential structures are there on a given topological manifold? In particular, why is the differential structure unique in dimensions $\leq 3$?
  • Which manifolds can be triangulated? In particular, why is it that all manifolds are triangulable in dimensions $\leq3$?

I'm looking for a textbook that addresses the above (and similar) questions, if such a book exists.

Depressingly, there is no textbooks covering any of these subjects. As far as I know, all of the research in low-dimensional topology can be done without knowing proofs of the existence/uniqueness of smooth structures on surfaces and 3-dimensional manifolds. In my estimation, less than 1% of the researchers in low-dimensional topology know these proofs. However, one does need to know the classification of 3-dimensional compact manifolds (and surfaces, of course). The classification was achieved by Perelman via Ricci Flow. This does not have a textbook treatment. (The book by Morgan and Tian comes closest but it only deals with manifolds with finite fundamental groups.) Almost all work in 3-dimensional topology is done without need to know the details of the proof. (There are few exceptions.)

As for the numbered questions, you really need an advisor to navigate in that area. Regarding

Is there any order in which I should study these topics? For example, should I study knots before 3-manifolds and vice versa? Should I study 3-manifolds before 4-manifolds?

I'd suggest to forget about "these topics:" Once you have an advisor, he/she will point you in the right direction which is aligned with his/her research interests. Maybe it will be 4-dimensional gauge theory, or knot invariants, or trisections of 4-manifolds, or hyperbolic geometry, etc. Most likely, you will read research papers, not books.

Edit. 1. For the record, there is a book

E.Moise, Geometric Topology in Dimensions 2 and 3, Springer Verlag, 1977

which proves existence and uniqueness of PL structure on topological 3-manifolds. However, it is very dated (it was dated by the time it was published) and I would not recommend it as a textbook in 3-dimensional topology. There are several nice introductory textbooks on 3-d topology, I will add a list later on.

  1. There is one book which can be used as a textbook, aimed at the classification of smooth manifolds of dimension $\ge 5$:

Kosinski, Antoni A., Differential manifolds, Pure and Applied Mathematics, 138. Boston, MA: Academic Press. xvi, 248 p. (1993). ZBL0767.57001.

  • this edition is expensive, but it was also republished in paperback form by Dover (2007) and is affordable.

It starts very gently (at the same level and pace as most textbooks on differential topology) but eventually gets to advanced topics such as the h-cobordism theorem and surgery theory.

One can also use Milnor's classic book "h-cobordism theorem," but it does not cover surgery theory and the learning curve in this book is much steeper.

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  • $\begingroup$ Thank you for your advice! To be honest, I'm quite surprised that few people really care about these proofs. Maybe it's just that I'm not yet used to take for granted theorems whose proofs I have never read about. $\endgroup$
    – Colescu
    Mar 9 '20 at 2:27
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    $\begingroup$ @Colescu: Life is short and one has to prioritize: Ultimately, you will be judged by what papers you wrote and what theorems you proved, not by what proofs of other people's theorems you know. (Unless you need these for your teaching or research.) That's life. $\endgroup$ Mar 9 '20 at 3:41

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