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Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over compact manifold $M$. If $M$ is not simply connected one can construct the universal covering $\widetilde{M}$ of $M$. If we do so, we have an action of $\pi_1(M)$ on $\widetilde{M}$.

I would like to understand in detail how we can lift $D$ to this universal covering in such a way that this operator will be equivariant.

If it is discussed in detail somewhere, I'll be satisfied with giving me some references (my quess is the following: first we have a covering map $p: \widetilde{M} \to M$ and then we can pull back the bundles $E,F$ to bundles $p^*(E),p^*(F)$: the action of $\pi_1(M)$ on $\widetilde{M}$ gives us the action on sections $(s \vartriangleleft g)(x)=s(g \vartriangleright x)$: but when we come to differential operator I don't know how to proceed further).

And once we have the lift $\widetilde{D}$ and we know that it is equivariant I read that one can associate to it a class in $K_0(C^*_r(\pi_1(M)))$. This is the 'higher index' of $D$.

I would like also to understand the main ingredients of this construction: how this higher index is defined, why it lives in $K$-theory of the reduced $C^*$-algebra of the group, do I need the equivariant $K$-theory to perform this construction?

Forgive me if my question is too broad: I'm starting my attempt to understand index theorem (the original one, due to Atiyah-Singer) but before this I would like to have some general picture about various generalisations (at least to know the formulation of theorems and understand constructions involved).

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A differential operator is locally defined, via charts or local coordinates. The projection $p:\tilde M\to M$ is a local diffeomorphism. So for each point $m\in\tilde M$ you have a neighborhood that maps diffeomorphically to a neighborhood of the image. So you can choose local coordinates around the point $m$ which come from local coordinates of $p(m)$. The expression of $D$ in these local coordinates gives you a local formula for the lift $\tilde D$. It is easily checked that two such local constructions give the same result where the open sets intersect, so you define the lifted operator unambiguously.

As for the second part of your question, the fundamental group $\pi_1$ acts on the kernel and cokernel of $\tilde D$ which defines representations of the $C^*$-algebra. Hence their difference gives an element in $K_0$.

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  • $\begingroup$ Let me just ask a few things: for the first, while everything is happening locally, then in suitably small neighborhood of $x \in \widetilde{M}$ the action is trivial therefore the lifted operator is invariant/equivariant? For the second part, could you elaborate a bit more how we get the element in $K_0(C^*_r(\Gamma)), \Gamma:=\pi_1(M)$ $\endgroup$ – truebaran Feb 9 '16 at 14:36
  • $\begingroup$ Look at the example $M={\mathbb R}/{\mathbb Z}$ and you see what's going on. $\endgroup$ – user1688 Feb 9 '16 at 17:48

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