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$$K$$-homology and index theory on contact manifolds. (English) Zbl 1323.58017
This paper studies the index problem for every Heisenberg-elliptic (pseudo-)differential operator $$P$$ on a closed contact manifold $$X$$. Such an operator is Fredholm, and one defines its index as the difference between the dimensions of the kernel and the cokernel of $$P$$.
The index formula, given first in terms of $$K$$-homology, involves two natural $$\mathrm{Spin}^c$$ structures on $$X$$, and two $$K^1(X)$$ classes $$\sigma^\pm_H(P)$$ extracted from the principal (Heisenberg) symbol of $$P$$.
Applying the homology Chern character to their formula, the authors deduce a cohomological index formula, involving the Chern of the symbol classes $$\sigma^\pm_H(P)$$ and the Todd class of the $$(1,0)$$, respectively $$(0,1)$$ components of the contact bundle $$H\to X$$.
The result in this paper generalizes previous index formulas due to L. Boutet de Monvel [Invent. Math. 50, 249–272 (1979; Zbl 0398.47018)], and C. Epstein and R. Melrose [“The Heisenberg algebra, index theory and homology”, Preprint].
In the case of scalar operators, the index formula was previously found by the second author [Ann. Math. (2) 171, No. 3, 1683–1706 (2010; Zbl 1206.19005)].

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 55N15 Topological $$K$$-theory 58J40 Pseudodifferential and Fourier integral operators on manifolds
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##### References:
 [1] Atiyah, M.F.; Singer, I.M., The index of elliptic operators. I., Ann. of Math.,, 87, 484-530, (1968) · Zbl 0164.24001 [2] Bargmann, V., On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math.,, 14, 187-214, (1961) · Zbl 0107.09102 [3] Baum, P. & Douglas, R.G., $$K$$ homology and index theory, in Operator Algebras and Applications(Kingston, Ont., 1980), Proc. Sympos. Pure Math., 38, Part I, pp. 117-173. Amer. Math. Soc., Providence, RI, 1982. [4] Baum, P. & Douglas, R.G., index theory, bordism, and $$K$$-homology, in Operator Algebras and K-theory (San Francisco, CA, 1981), Contemp. Math., 10, pp. 1-31. Amer. Math. Soc., Providence, RI, 1982. [5] Baum, P.; Higson, N.; Schick, T., On the equivalence of geometric and analytic K-homology, Pure Appl. Math. Q.,, 3, 1-24, (2007) · Zbl 1146.19004 [6] Beals, R. & Greiner, P., Calculus on Heisenberg Manifolds. Annals of Mathematics Studies, 119. Princeton University Press, Princeton, NJ, 1988. · Zbl 0654.58033 [7] Blackadar, B., $$K$$-Theory for Operator Algebras. Mathematical Sciences Research Institute Publications, 5. Cambridge University Press, Cambridge, 1998. [8] Boutet de Monvel, L., On the index of Toeplitz operators of several complex variables. Invent. Math., 50 (1978/79), 249-272. · Zbl 0398.47018 [9] Choi, M.D.; Effros, E.G., The completely positive lifting problem for $$C$$*-algebras, Ann. of Math.,, 104, 585-609, (1976) · Zbl 0361.46067 [10] Connes, A., Noncommutative Geometry. Academic Press, San Diego, CA, 1994. [11] Connes, A.; Moscovici, H., The local index formula in noncommutative geometry, Geom. Funct. Anal.,, 5, 174-243, (1995) · Zbl 0960.46048 [12] Connes, A.; Moscovici, H., Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys.,, 198, 199-246, (1998) · Zbl 0940.58005 [13] Epstein, C. L., Lectures on indices and relative indices on contact and CR-manifolds, in Woods Hole Mathematics, Ser. Knots Everything, 34, pp. 27-93. World Scientific, Hackensack, NJ, 2004. [14] Epstein, C. L. & Melrose, R., The Heisenberg algebra, index theory and homology. Preprint, 2004. [15] Erp, E., The Atiyah-Singer index formula for subelliptic operators on contact manifolds. part I, Ann. of Math.,, 171, 1647-1681, (2010) · Zbl 1206.19004 [16] van Erp, E., The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part II. Ann. of Math., 171 (2010), 1683-1706. · Zbl 1206.19005 [17] Erp, E., Noncommutative topology and the world’s simplest index theorem, Proc. Natl. Acad. Sci. USA,, 107, 8549-8556, (2010) · Zbl 1205.58006 [18] van Erp, E., The intrinsic geometry of the osculating structures that underlie the Heisenberg calculus. Preprint, 2010. arXiv:1007.4759 [math.AP]. [19] Folland, G. B. & Stein, E. M., Estimates for the $${\bar{∂}_b}$$ complex and analysis on the Heisenberg group. Comm. Pure Appl. Math., 27 (1974), 429-522. · Zbl 0293.35012 [20] Melrose, R., Homology and the Heisenberg algebra, in Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, Exp. No. XII. École Polytechnique, Palaiseau, 1997. [21] Taylor, M. E., Noncommutative microlocal analysis. I. Mem. Amer. Math. Soc., 52 (1984).
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