zbMATH — the first resource for mathematics

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.

58J22 Exotic index theories on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
Full Text: DOI
[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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.