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Can anyone help me to find a simple and good reference (a book, lecture notes or a website) for learning the surgery theory and its applications? I seek a reference together with many examples and figures for more intuition.


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Well, there's surgery and there's Surgery Theory. By the former, I mean the basic geometric and homotopy-theoretic ideas that went into building a big machine (the latter) to systematically study problems in high-dimensional manifolds. Although you can dive right into Surgery Theory, I think it's much better to invest some time in learning the basic ideas, from the original paper of Milnor (A procedure for killing homotopy groups of differentiable manifolds. 1961 Proc. Sympos. Pure Math., Vol. III pp. 39–55). The introduction to Ranicki's book referenced above makes the same recommendation. The basic `example' is the application of surgery to classifying smooth structures on homotopy spheres (Kervaire-Milnor, Groups of homotopy spheres. I. Ann. of Math. (2) 77 1963 504–537.)

After that, it may be reasonable to start learning the more systematic version, which was formulated by Browder-Novikov-Wall (and many others). Ranicki's book is a good place to start with this, but if you really want to understand examples, you will at some point need to read Wall's book on non-simply-connected surgery.

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Sorry to disappoint you but there is no reference with many figures and examples, and for a good reason: Surgery theory works best in higher dimensions so most pictures are schematic at best, while examples are notoriously hard to work out, and there very few of them in textbooks (and not that many in the literature). Also surgery is at its best analysing implicit and partial information, so it is unreasonable to hope for lots of pictures and examples.

I suggest the following road map:

  1. Homotopy theory (as e.g. in May's textbook), characteristic classes (Milnor-Stasheff), h-cobordism theorem (Milnor and Kosinski's "Differential manifolds").

  2. Papers by Milnor and Kervaire-Milnor (see Danny Ruberman's answer).

  3. Browder's book "Surgery on simply-connected manifolds" and Ranicki's "Algebraic and Geometric Surgery".

  4. Wall's "Surgery theory" and Lueck's "A Basic Introduction to Surgery Theory".

  5. Orginal sources: a huge list is being maintained by Andrew Ranicki.

Disclaimer: I am no expert in surgery theory, but I have been studying it for many years and finally got to the point of using it for Riemannian geometry purposes.

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Scorpan's "the wild world of 4-manifold" is, as its title indicate, restricted to dimension 4 but it has far more picture than one would think possible, including about some surgery constructions. The book is nicely written, and combine an overview of the subject with non-mandatory, small font technical details.

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