So far I made several attempts to really learn Atiyah-Singer theorem. In order 
to really understand this result a rather broad background is required: you need
to know analysis (pseudodifferential operators), algebra (Clifford algebras, 
spin groups) and algebraic topology (characteristic classes, K-theory, Chern 
character). So every time I have stuck in some place and have too many doubts to 
continue my journey into the realm of index theory. Finally I came to conclusion that what I'm lacking is some big picture about various approaches to index theorem: there are many (excellent) books that treat the subject but before you arrive at the proof of index theorem itself you have to go through about 150 pages of background material (which usually you *partially* know). 


**Some issues which I already figured
out**:


1. There are at least two approaches: the one which uses K-theory and the second
so called heat kernel approach. As I understood for the first one good knowledge of algebraic topology is required while for the second you have to know differential geometry stuff (connections, curvatures etc.)
2. As far as I know these approaches have different range of applicability: the
heat kernel approach is more restrictive and do not include all differential elliptic operators.
3. There are some other proofs (using K homology - see the book of Higson and Roe, via tangent groupoid, there is also quite recent proof due to Paul Baum)


So what answer(s) I would be very pleased to see? 



> 
> 
> > 
> > I would like to gather in one place 
> >  various possible approaches to the proof of index theorem together with discussion of
> >  the main ingredients needed to complete such proof (for example: "you need good
> >  understanding of K-theory not only for compact spaces, but also for locally compact. 
> >  You also have to understand Bott periodicity and Thom isomorphism theorem), the 
> >  formulation of the index formula together with comparison and relation to other 
> >  index formula (for example: you obtain topological index as a K-theory class- 
> >  in order to obtain the famous cohomological formula (involving Todd class) you apply 
> >  Chern character) and finally the level of generality of formulation of index theorem
> >  (for example: this approach works for all elliptic differential operators). 
> > 
> > 
> > 
> 
> 
> 


Maybe this question is broad but I have an impression that I'm not the only person who
will find such a discussion useful. 



Atiyah-Singer theorem-a big picture
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](https://arxiv.org/pdf/1604.03535.pdf) 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](https://arxiv.org/pdf/1604.03502.pdf) 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.


