# Second Steps in Topology.

There are many possibilities. Here are some. I would probably recommend that you look at all of them -- mostly these all develop techniques that are useful in many directions within and without topology.

1. Characteristic classes.

These are invariants of vector bundles and provide a vast extension of the Poincare-Hopf fact that the alternating sum of the indices of a vector field on a manifold is independent of the choice of vector field.

Characteristic classes can be introduced geometrically, differential geometically and purely algebraic topologically (and in more than one way). That there are so many viewpoints underscores their fundamental nature and is also a source of their utility. The canonical reference is Milnor and Stasheff's "Characteristic Classes".

2. Morse theory.

Morse theory studies the interaction between the singularities of real valued smooth functions and the topology of the space they are defined on. In many situations it leads to elegant ways to compute homology; it is also a way into handlebody theory, the basic discretization of the topology of manifolds. (Why are there only countably many diffeomorphism classes of compact smooth manifolds?)

The best introductory reference is Milnor's "Morse Theory", which includes a very useful rapid course in differential geometry and the proof of Bott periodicity.

For the handlebody picture of manifolds, and its first and most important consequence, the h-cobordism theorem, see Milnor's "The h-cobordism theorem", also Rourke and Sanderson's "Introduction to PL topology" and the paper "Le theoreme de Barden-Mazur-Stallings" by Kervaire in Comm. Math. Helv. (The latter is also good practice for your French exam.)

3. K-theory

This is the systematic analysis of vector bundles. Of course, as a subject, it interacts with characteristic classes, but here the focus is on the aggregate of all vector bundles, not on the individual ones.

Topological K-theory really began with Bott's periodicity theorem - one proof of which comes via Morse theory. There must be more than a half dozen different proofs known, by now.

There are at least three different aspects of the subject that the beginner will want to learn.

The first is essentially formal and deals with K-theory as a generalized cohomology theory, and what that entails (e.g. some integrality theorems for certain combinations of characteteristic classes and the Atiyah-Hirzebruch spectral sequence.)

The second is the role of Adams operations. These are built out the exterior powers, and are a powerful bit of extra structure that K-theory has. It leads to very deep results, such as the nonparallelizability of all the spheres besides those of dimensions 1,3, and 7, the fact that besides the 2 and 6 spheres, no sphere has a complex structure, and the existence of many nontrivial elements in the stable homotopy groups of spheres (the "image of J").

Finally, there is the intimate connection between K-theory and elliptic operators (and the index theorem).

Proabably the best thing to do is to peruse the middle three volumes of Atiyah's collected works (beginning with his text "K-theory" and learning the elliptic operator proof of Bott periodicity in place of the "elementary proof" that's presented in his book.) Another fine reference is Lawson and Michelson's book "Spin geometry" that focuses on the connections to the index theorem.

4. Differential geometry.

The first year course somehow doesn't catch much of the spirit of the subject. I would recommend Cheeger and Ebin's beautiful book on "Comparison differential geometry" as a classic. (Don't forget Milnor's efficient "rapid course" mentioned above in Morse theory") There are more recent books, Hulin et al, Chavel, and Petersen, that are all well written and emphasize different points of view.

5. Three-manifolds

One can learn a good deal of the theory without an enromous amount of "post transversality" topology. In my (eccentric) opinion a good beginning could be Rolfsen's book, "Knots and links" and Jaco's or Hempel's books on three manifolds.

All of this will only take one to the "pre-geometrization era" which will then require one to learn about hyperbolic geometry, foliations, laminations, Teichmuller theory, etc. etc.

(Another direction one coud go after Rolfsen is into quantum invariants of knots and three-manifolds, which is an entirely different matter.)

6. Algebraic K-theory

Here the main introductory topics are the Wall finiteness obstruction (how can one tell whether or not a CW complex is homotopy equivalent to a finite complex) and simple homotopy theory (i.e. the theory of "simple homotopies") which enters in showing that the homotopy equivalent lens spaces are diffeomerphic iff they obviously are (i.e. it gives the first source of manifolds that are homotopy equivalent and non-diffeomorphic; these examples are all parallelizable in dimension three, so one can't distinguish these by characteristic classes).

The best first references are Cohen's "Course in simple homotopy theory", Milnor's 1965 Bull AMS article on "Whitehead torsion", Wall's Annals of Math paper "Finiteness conditions for CW complexes".

Next Steps: Immersion theory, Surgery theory, Controlled topology, higher simple homotopy theory, the index theorem and its extensions, 4-manifolds, symplectic geometry, harmonic maps, Mostow rigidity, superrigidity, geomteric group theory etc.

After that come the subjects that you will create. Have fun!