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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)].

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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