# Role of Dirac operators in Index Theorems

I'm trying to approach the Atiyah-Singer Index Theorem by getting an overview of the area.

One thing that confuses me a lot is that some treatments give (and hence prove) the theorem for Dirac operators, while other sources not even mention Dirac operators and work just with general elliptic operators.

From what I read seems that the Index theorem for Dirac operators implies the theorem for general elliptic operators. Indeed the Dirac operator is some sense "the" elliptic operator. I've heard this claim can be justified with some K-theory (which I know nothing about at the moment).

Can someone explain in simple (i.e. vague) terms this idea? Why the Dirac operator is "the" elliptic operator?

Apart from the K-theoretical justification, are there any methods of showing this? Can this be seen from the Analysis side?

What this boils down to is that each symbol determines a class in a certain abelian group, which is defined topologically (and the whole group is obtained in that way). For all symbols in a class, the corresponding operators have the same index, so this defines a homomorphism from that group to $$\mathbb Z$$ (usually called the "analytical index"). Using pure topology (characteristic classes, Chern character,etc.), one defines another homomorphism (calles the "topological index") between the same groups and the index theorem boils down to showing that these two homomorphisms agree.