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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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References:
[1] A. Connes, Noncommutative Geometry (Academic, San Diego, 1994).
[2] A. Antonevich and A. Lebedev, Functional Differential Equations, Vol. 1: C*-Theory (Longman, Harlow, 1994). · Zbl 0799.34001
[3] A. Antonevich and A. Lebedev, Functional Differential Equations, Vol. 2: C*-Applications, Part 1: Equations with Continuous Coefficients (Longman, Harlow, 1998). · Zbl 0936.35207
[4] V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, Elliptic Theory and Noncommutative Geometry (Birkhauser, Basel, 2008). · Zbl 1158.58013
[5] A. Yu. Savin and B. Yu. Sternin, Dokl. Math. 82, 519–522 (2010) [Dokl. Akad. Nauk 433, 21–24 (2010)]. · Zbl 1213.58016 · doi:10.1134/S1064562410040058
[6] A. Yu. Savin, Dokl. Math. 82, 884–886 (2010) [Dokl. Akad. Nauk 432, 170–172 (2010)]. · Zbl 1235.58018 · doi:10.1134/S1064562410060128
[7] D. P. Williams, Crossed Products of C*-Algebras (Am. Math. Soc., Providence, RI, 2007). · Zbl 1119.46002
[8] A. Haefliger, J. Differ. Geom. 15, 269–284 (1980). · Zbl 0444.57016 · doi:10.4310/jdg/1214435494
[9] A. Yu. Savin and B. Yu. Sternin, Mat. Sb. 201(3), 63–106 (2010). · doi:10.4213/sm7537
[10] A. Savin, K-Theory 34(1), 71–98 (2005). · Zbl 1087.58013 · doi:10.1007/s10977-005-1515-1
[11] G. de Rham, Variétès différentiables (Hermann, Paris, 1955).
[12] M. F. Atiyah and I. M. Singer, Bull. Am. Math. Soc. 69, 422–433 (1963). · Zbl 0118.31203 · doi:10.1090/S0002-9904-1963-10957-X
[13] N. Kryloff and N. Bogoliuboff, Ann. Math. 38(1), 65–113 (1937). · Zbl 0016.08604 · doi:10.2307/1968511
[14] D. V. Anosov, Usp. Mat. Nauk 49(5), 5–20 (1994).
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