K-theory formulation of the index theorem

Question

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold $M$ one has $\operatorname{Ind}D=\langle\operatorname{ch}(D) \cdot\operatorname{Td}(M), [M] \rangle$. This is the form of index theorem which I know: however, reading some expository texts I met the statement that it is nice to translate $K$-theoretic formulation of the index theorem into the “local” form where characteristic classes are involved. I wonder what does it mean:

Question What is the $K$-theoretic formulation of the index theorem?

I would be also very grateful for some historical explanation about the issues of $K$-theoretical formulation of index theorem and the one which I formulated (which one was earlier? all proofs are due to Atiyah and Singer? Which of them is seen to be more improtant? and so on...)

Answer 1

From the Wikipedia page on the index theorem:

One can also define the topological index using only K theory (and this alternative definition is compatible in a certain sense with the Chern-character construction above). If $X$ is a compact submanifold of a manifold $Y$ then there is a pushforward (or "shriek") map from $K(TX)$ to $K(TY)$. The topological index of an element of $K(TX)$ is defined to be the image of this operation with $Y$ some Euclidean space, for which $K(TY)$ can be naturally identified with the integers $\mathbb Z$ (as a consequence of Bott-periodicity). This map is independent of the embedding of $X$ in Euclidean space. Now a differential operator as above naturally defines an element of $K(TX)$, and the image in $\mathbb Z$ under this map "is" the topological index.

The Wikipedia article is very detailed and answers pretty much all your question. The proof due to Atiyah--Singer and uses K-theory. The first proof used the Hirzebruch–Riemann–Roch theorem (I'm just quoting excerpts from Wikipedia at this point). Other proofs were given by other people...

Answer 2

I disagree that index theorem is "one of the most important results in twentieth century's mathematics". This kind of superlatives usually does not mean anything, unless one can draw a fine line between "important", "more important", "most important", "of greatest importance", etc. I think it is better to be humble about it, even though my own research field is index theory and I wrote a thesis on it. Personally I think Atiyah-Singer is remarkable, but definitely not "one of the most important"results I know.

Having said that, I encourage you read an expository article on the K-theory proof of index theorem. The statement of theorem in K-theory setting is in page 17. The central part of the K-theory proof of index theorem is based on Thom isomorphism, and its proof in K-theory setting is not easy. So maybe a starting point is to go through Atiyah's book K-theory and Hatcher's book Vector bundles and K-theory, then going back to the proof itself.

You asked "I met the statement that it is nice to translate K-theoretic formulation of the index theorem into the “local” form where characteristic classes are involved. ". This is (in some sense) incorrect, and you should check the original source of this statement. The K-theory proof of the index theorem is global and not easily computable, what this statement is alluding to is the heat kernel proof of index theorem, which is usually called "the local index theorem" as it gives the topological side as an integral of differential forms in a local form: $$ Ind(\mathcal{D})={\frac{1}{(4\pi i)^{m}}\int_{M}(\det)^{1/2}(\frac{\mathcal{R}/2}{\sinh(\mathcal{R}/2)})Tr(\tilde{Z}e^{\tilde{Q}})} $$ But this statement has almost no characteristic class involved, what I mean is you can think of it purely in terms of differential forms without need to appeal to definition of Chern character or Todd class. The heat-kernel proof has the benefit of being computable, for example one can use it to show Atiyah-Singer implies Gauss-Bonnet and Riemann-Roch relatively easily. However, the statement is somehow uglier and which one you prefer is matter of aesthetic. Since the heat-kernel proof is local in nature, it is not clear to how translate it into an appropriate algebraic setting (like homogeneous spaces, loop spaces, etc) where one may expect a simplified version of Atiyah-Singer. And similarly K-theory proof does not work well when one wants to use it to make computation in an arbitrary given manifold.

There are also other proofs, using cobordism theory(the original proof), probability theory, and even path integral. But it is more or less outside of my expertise as I never read these proofs carefully. So I cannot comment on it.