K-theory formulation of the index theorem 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...)

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...
