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Here is a very rough outline of the proof of the index theorem using KK-theory: Define $KK_G(A, B)$, where $G$ is a Lie group and $A$ and $B$ are [adjectives] C*-algebras, and the Kasparov product be …

The assertion is supposed to be that $d(e^{-tD^2})/dt$ has the same smooth kernel as $-D^2 e^{-tD^2}$, i.e. they are the same operator. This is because $e^{-tD^2}$ is the solution operator to the hea …

There are a number of ways to do this calculation, but at risk of shamelessly plugging my own work there is a nice way to see it using a Mayer-Vietoris principle. Decompose $\mathbb{R}$ as the union …

Perhaps the simplest answer is to use the spectral theorem: $L^2(S)$ decomposes as the orthogonal direct sum of $D$-eigenspaces, and $f(D)$ acts on each $\lambda$-eigenspace as multiplication by $f(\l …

What little background I have in this area is probably outdated, but I can share a few thoughts. The "index theorem" to which Hopkins was most likely referring was in Witten's 1987 paper The Index of …

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 var …

Perhaps you would be interested in Witten's proof of the Poincare-Hopf theorem. Given a smooth nondegenerate vector field $V$ on a smooth closed manifold $M$, the theorem asserts that the Euler chara …

Index theory is fundamentally about a homomorphism $$K_n(M) \to \mathbb{Z}$$ from the top degree K-homology of $M$ (even dimensional) to the integers called the analytic index map. It is called this …

This isn't really going to be an answer, but it's too long to be a comment and I think it will be helpful. First, as the other answerers pointed out it is a good idea to avoid boundary value problems …

1) The eta invariant itself depends on the metric, but the relative eta invariant is in many cases (see comments) a homotopy invariant. The relative eta invariant is defined to be the difference of t …

I think that some modest bit of equivariant K-theory is required in the K-theory proof of the index theorem. You correctly identified the main place where it enters into IEO1, but I think there is a …

I can't give you an answer that is well-adapted to Lawson and Michelsohn's formulation of the pseudodifferential calculus. But here's how this sort of argument is supposed to go: two elliptic pseudod …

I don't think I can really give you the intuition that you seek because I don't think I quite have it yet either. But I think that understanding the relevance of Nigel Higson's comment might help, an …