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.
- Would a book entirely devoted to, say, $3$-manifolds, be too specific for what I want to know?
- Which topics (among, e.g., surgery theory, PL topology, knot theory, $4$-manifolds) are relevant here?
- 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?
- Could you please suggest an (ordered) list of books to read?
- 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!