Atiyah-Singer theorem-a big picture

Well, I guess that there is no royal road to the Index Theorem.

I think that what is needed in order to understand the Atyiah-Singer index theorem is the opposite of a big picture. It is easier to understand particular cases before jumping to the full statement. The big problem if you dive right into some proof of the Index Theorem is that it won't provide any examples to build intuition and understanding. That's really an abstraction over abstractions over geometric and analytical objects.

So I would suggest to start with the applications of the theorem, understand these particular cases on their own, before moving to the general proof. By reading I.2 and chapter II of the book of P. Shanahan entitled "The Atyiah-Singer Index theorem", LNM 638, even if not everything makes sense, you will get a "big picture", so to speak, of the applications of the Index Theorem and then you can start diving into the classical proofs of these applications, which will teach you in order, one at a time, all the different notions you need to understand the Index Theorem. Use chapter II of Shanahan's book as a roadmap.

The Atiyah-Singer theorem is really a synthesis of many theorems from many parts of mathematics. I suggest some references above, but really you just need to browse through them, not read them from front to cover. So, here are some suggestion for self-study.

The Euler characteristic

The first application of the Index theorem is to recover the Gauss-Bonnet formula on surfaces, which establishes a relationship between the Euler characteristic of a surface and the integral of the curvature of some Riemannian metric on the surface. and the Hopf theorem connecting the Euler characteristic with the zeros of vector fields. By learning the classical proofs of these results (e.g. in Do Carmo "curves and surfaces" and perhaps Milnor, "topology from the differential viewpoint"), you will see interesting examples of the interplay between topology and differential geometry.

It is then good to read more about the Euler characteristic in an algebraic topology book (e.g. Greenberg, "algebraic topology" Part 2, ch 20) and a differential topology book (e.g. Hirsch "differential topology", ch 4 and 5 or Bott-Tu "differential forms in algebraic topology" I.6) You then need to learn about the relationship between the Euler characteristic and the De Rham cohomology of differential forms. The Laplacian makes its first appearance there. I am not sure what to recommend here, perhaps a book on pseudo-differential operators, or ch I and III of Rosenberg "The Laplacian on a Riemannan Manifold", where the Gauss-Bonnet theorem is again discussed.

The Euler characteristic is a particular case of a characteristic class, so you are probably ready at that point to read a large part of the book of Milnor "Characteristic classes" or the relevant part of Bott-Tu.

The Lefschetz fixed point theorem

This fixed point theorem is yet another special case of the Index Theorem. You can learn about it in e.g. Greenberg and make the connection with the differential approach in Hirsch and Bott-Tu. It is also interesting to have a look at the more combinatorial approach that can be found in the original book of Lefschetz.

The Riemann-Roch-Hirzebruch theorem

This formula appears in analytic geometry and computes the dimension spaces of meromorphic functions with value in holomorphic vector bundles. The case of curves (Riemann surfaces) is quite interesting to study. A new operator, the Dolbeaut operator, makes its appearance here. Perhaps look at Griffiths-Harris, "principles of algebraic geometry", ch.II, III.3, III.4. but there should be lighter books devoted to the case of Riemann-Roch for surfaces using the Dolbeaut operator.

At that point, you should see that there is a common denominator in all these results and then it is time to look at the general proof of the Index Theorem.

Just for completeness let me add a few words on the heat kernel proof. It is true that some analysis is needed. But one should not forget that even the $K$-theoretic proof depends on the notion of pseudo-differential operators and therefore cannot avoid analysis completely.

It is also true that the proof really only works well for Dirac operators. But then, Dirac operators generate $K$-homology, and most applications of the index theorem are formulated in terms of (0-order perturbations of) Dirac operators. Contradicting comments are welcome! The most prominent examples are mentioned in Coudy's answer, but one should add the signature operator (which is the Euler operator with respect to a different grading), and the so-called untwisted Dirac operator.

The only real drawback is that one only sees the image of the index in rational cohomology. But then, using family index techniques or $\eta$-invariants, one can sometimes recover a bit more of the topological content of the index.

The proof consists of the following main ideas.

1. The McKean-Singer trick. This still works for all elliptic differential operators. If $P\colon\Gamma(E)\to\Gamma(F)$ is an elliptic differential operator on a manifold $M$, consider its adjoint $P^*\colon\Gamma(F)\to\Gamma(E)$. Then it makes sense to consider $$\operatorname{ind}(P)=\operatorname{tr}\bigl(e^{-tP^*P}\bigr)- \operatorname{tr}\bigl(e^{-tPP^*}\bigr)\;.$$ In the limit $t\to 0$, the traces above become local, that is, they can be represented by an integral over $M$ that depends only on the coefficients of $P$ in local coordinates.

From here on, one could try to prove that only finitely many geometric terms can contribute to the index. This was the idea of Gilkey and Patodi. One difficulty comes from the fact that the relevant contribution to the index sits in the $t^0$-term of the asymptotic expansion, which is by far not the leading term. However, in the special case of Dirac operators, one gets around this problem.

2. Getzler rescaling. If $P=D$ is a Dirac operator, then $\Delta=D^*D\oplus DD^*$ is a Laplace operator, and $e^{-t\Delta}$ is the standard heat kernel. One performs a standard parabolic rescaling of space and time together with Getzler's rescaling of the Clifford algebra to show that as $t\to 0$, $\Delta$ converges against a model operator that looks like a harmonic oscillator with coefficients in $\Lambda^{\mathrm{even}}T^*M$. The relevant contribution to the index now appears in the leading order term thanks to Getzler's trick. This only works if $D$ is compatible with a Riemannian metric (here one makes use of the homotopy invariance of the index to achieve this), and one has to chose nice local coordinates.

3. Mehler's formula. The heat kernel for the model operator has an easy explicit formula in terms of the Riemannian curvature and the coefficients of the Dirac operator.

4. Chern-Weil theory. The output of step 3 is an integral over $M$ of an invariant polynomial, applied to the Riemannian curvature and the so-called twisting curvature of the Dirac bundle on which $D$ acts. Hence it can be interpreted as the evaluation on the fundamental cycle of $M$ of an expression in characteristic classes.

Indeed, as pointed out in the comments to Paul's answer, one does not need any advanced analysis. Some notions from a first course in Riemannian geometry are needed. The most tricky part seems to be Getzler's rescaling, which is mostly algebraic.

I agree with @coudy's answer that the best approach is to first understand the theorem's special cases / applications / generalizations. That can help highlight some of the key pain points in the various proofs, and motivate some of the ideas involved. Still, I'll take a crack at the main thrust of the question: how do the proofs work and what's involved?

I think basically all of the proofs can be organized into three categories:

1. K-theory (topology)
2. K-theory (operator algebras)
3. Heat kernels

1 and 2 are fairly similar and probably more or less equivalent, but they lend themselves to different generalizations. The techniques of 2 are responsible for many of the most state-of-the-art applications, e.g. to the Novikov conjecture or to noncommutative geometry.

3 seems to be completely different, or at least I don't think anybody can claim to understand why the techniques in 1 and 2 are capable of proving the same theorem as the techniques in 3. On the other hand, 3 is required (given the current state of the literature) for certain applications and generalizations, such as the Atiyah-Patodi-Singer index theorem for manifolds with boundary. It's also quite hard to summarize the main ideas - a lot of gritty analysis and PDE theory is involved.

For this answer I'll try to explain and compare 1 and 2; if I have the time later I might revisit 3 in another answer.

K-Theory Proofs

Both types of K-theory proofs (1 and 2) follow the same basic pattern; the differences are in how the relevant maps are defined and computed. Here's a general schema expressed in the modern way of thinking (emphasizing K-homology and Dirac operators).

1. Define the Dirac operator $D$ on a Riemannian spin (or spin$^c$) manifold $M^{2k}$, and show that it defines the fundamental class in K-homology: $[D] \in K_{2k}(M) \cong K_0(M)$.
2. Show that the Poincare duality pairing between K-theory and K-homology applied to the fundamental class $[D]$ sends the K-theory class of a vector bundle $E$ to the Fredholm index of the twisted Dirac operator $D_E$. This gives an analytic index map $K^0(M) \to \mathbb{Z}$ which is an isomorphism.
3. Construct a topological index map $K^0(M) \to \mathbb{Z}$ as follows. Embed $M$ into $\mathbb{R}^n$, apply the Thom isomorphism to get a K-theory class on the normal bundle of $M$, use a diffeomorphism from the normal bundle to an open set in $\mathbb{R}^n$ to get a K-theory class on an open set in $\mathbb{R}^n$, and apply the wrong-way map in K-theory to get a class in $K_0(\mathbb{R}^n) \cong \mathbb{Z}$.
4. Calculate that the analytic and topological index maps agree.
5. Apply Chern characters everywhere to get a formula in cohomology - note that the Thom isomorphism in K-theory and the Thom isomorphism in cohomology are not compatible with Chern characters, so the Todd class appears as a correction term.
6. Reduce the index theorem for an arbitrary elliptic operator to the index theorem for spinor Dirac operators. (This is some version of the clutching construction; here is a good modern reference.)

This should indicate what the prerequisites are: a little spin geometry to define Dirac operators, some analysis to show that the Fredholm index exists and is well-defined on K-theory, and some topology to construct the topological index map.

Proof 1 (Topological K-theory)

The strategy of the proof is to show that the analytic index map is an isomorphism, the topological index map is a homomorphism, and both maps are functorial in $M$. This means that the two maps are always equal if they are equal on one example, and one can check by direct calculation on, say, the sphere (where the index theorem is basically just the Bott periodicity theorem).

A good reference is Atiyah and Singer's original paper "Index of Elliptic Operators I", though it should be noted that they don't explicitly use K-homology and neither Dirac operators nor the cohomologlical formula are introduced until IEO III. Nevertheless, the ideas are pretty much the same.

Baum and Van Erp provide a modern reference which fills out the schema using purely topological methods.

Proof 2 (Operator K-theory)

The idea of the operator algebraic proof is to use Kasparov's bivariant groups $KK(A,B)$ where $A$ and $B$ are C*-algebras. The K-homology of $M$ is the special case $KK(C(M), \mathbb{C})$ and the K-theory is the special case $KK(\mathbb{C}, C(M))$. There is a product in KK-theory:

$$KK(A,B) \times KK(B,C) \to KK(A,C)$$

and in the special case where $A = C = \mathbb{C}$ and $B = C(M)$ one recovers the analytic index map as:

$$K^0(M) \times K_0(M) \to KK_0(\mathbb{C}, \mathbb{C}) \cong \mathbb{Z}$$

(i.e. the product of the K-theory class of a vector bundle and the K-homology class of the Dirac operator is the index of the operator twisted by the bundle.) The KK product is functorial for C*-algebra homomorphisms in all factors, and it is compatible with all of the ingredients in the topological index map (e.g. the Thom isomorphism is just a KK product with the Bott element in K-theory). So the proof of the index theorem becomes a simple little calculation with KK products.

This is a very powerful and appealing approach, but the Kasparov groups and especially the Kasparov product are hard to define. Probably the best references are K-theory for C$^\ast$ algebras by Blackadar and Analytic K-homology by Higson and Roe.