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An index formula for nonlocal operators corresponding to a diffeomorphism of a manifold. (English. Russian original) Zbl 1246.58017
Dokl. Math. 83, No. 3, 353-356 (2011); translation from Dokl. Akad. Nauk 438, No. 4, 444-447 (2011).
Let $$M$$ be an oriented smooth manifold, let $$g:M\to M$$ be an orientation preserving diffeomorphism, and let $$E$$ and $$F$$ be complex vector bundles on $$M$$.
The authors consider nonlocal operators given by a finite sum $$D=\sum_kD_kT^k:C^\infty(M,E)\to C^\infty(M,F)$$, where $$T:C^\infty(M,E)\to C^\infty(M,g^*F)$$ is the shift operator induced by $$g$$, $$(Tu)(x)=u(g(x))$$, and the coefficients $$D_k:C^\infty(M,g^{k*}E)\to C^\infty(M,F)$$ are zero order pseudodifferential operators. The symbol $$\sigma(D)$$ is the sequence of symbols $$\sigma(D_k)\in C^\infty(S^*M,\pi^*\text{Hom}(g^{k*}E,F))$$, where $$\pi:S^*M=T_0^*M/\mathbb{R}_+\to M$$ is the cospherical bundle. If $$B=\sum_kB_kT^k:C^\infty(M,G)\to C^\infty(M,E)$$ is another operator of the same type, then $$\sigma(DB)(k)=\sum_{\ell+m=k}\sigma(D_\ell)((\partial g^\ell)^*\sigma(B_m))$$, where $$\partial g=(dg^t)^{-1}:T^*M\to T^*M$$. This formula is taken as definition of product of this kind of symbols, and the symbol of the identity operator $$T^0$$ is the identity element. Then it is said that $$D$$ is elliptic if $$\sigma(D)$$ is invertible. In this case, it is observed that $$D$$ defines a Fredholm operator between Sobolev spaces of any order, and its kernel and cokernel consist of smooth sections.
In [$$K$$-Theory 34, No. 1, 71–98 (2005; Zbl 1087.58013)], the first author defined the difference construction $$[\sigma(D)]\in K_0(C^\infty_0(\mathcal{T}^*M)\rtimes\mathbb{Z})$$, where $$C^\infty_0(\mathcal{T}^*M)$$ is the algebra consisting of all $$u\in C^\infty([0,1]\times S^*M)$$ such that $$u|_{t>1-\epsilon}=0$$ and $$u|_{t<\epsilon}=u_0(x)$$ for $$u_0\in C^\infty(M)$$. Similarly, let $$\Lambda(\mathcal{T}^*M)$$ be the differential algebra consisting of all $$\omega$$ in the de Rham differential algebra $$\Lambda[0,1]\times S^*M)$$ so that $$\omega|_{t>1-\epsilon}=0$$ and $$\omega|_{t<\epsilon}=\pi^*\omega_0$$, where $$\pi:[0,1]\times S^*M\to M$$ is the projection. Consider the corresponding Haefliger cohomology $$H^*(\mathcal{T}^*M/\mathbb{Z})$$, defined by the complex $$\Lambda(\mathcal{T}^*M)/(1-g^*)\Lambda(\mathcal{T}^*M)$$. Then the Chern character $$\text{ch}:K_*(\mathcal{T}^*M)\rtimes\mathbb{Z})\to H^*(\mathcal{T}^*M/\mathbb{Z})$$ is defined. Moreover $$\pi:[0,1]\times S^*M\to M$$ induces a homomorphism $$\pi_*:H^*(\mathcal{T}^*M/\mathbb{Z})\to H^{*-\dim M}(M/\mathbb{Z})$$, where $$H^*(M/\mathbb{Z})$$ is the Haefliger cohomology defined by $$(M,g)$$ (given by the complex $$\Lambda(M)/(1-g^*)\Lambda(M)$$). The Chern character of an elliptic $$D$$ is defined as $$\text{ch}(D)=\pi_*\cosh[\sigma(D)]\in H^*(M/\mathbb{Z})$$.
On the other hand, let $$H_*(M/\mathbb{Z})$$ be the homology of the complex of $$g$$-invariant currents on $$M$$, which has a canonical pairing with $$H^*(M/\mathbb{Z})$$. The authors assume that the class $$\text{Td}(T^*M\otimes\mathbb{C})\cap[M]\in H_*(M)$$, dual to $$\text{Td}(T^*M\otimes\mathbb{C})$$, belongs to the image of the canonical homomorphism $$H_*(M/\mathbb{Z})\to H_*(M)$$, and take a preimage denoted by $$\text{Td}_g(T^*M\otimes\mathbb{C})$$. Then the topological index of an elliptic $$D$$ is defined as $$\text{ind}_tD=\langle\text{ch}D,\text{Td}_g(T^*M\otimes\mathbb{C})\rangle$$. The main result of the paper is an index theorem stating that, if $$H_{\text{odd}}(M)\otimes\mathbb{Q}=0$$, then $$\text{ind}D=\text{ind}_tD$$. The proof is not given, but several examples are described.

##### MSC:
 58J22 Exotic index theories on manifolds 58J20 Index theory and related fixed-point theorems on manifolds
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