In this paper we give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural {\mathbb{R}^{\times}_{+}}-action. Specifically, a properly supported semiregular distribution on {M\times M} is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibers of the tangent groupoid that is homogeneous for the {\mathbb{R}^{\times}_{+}}-action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.

The problem and the solution 2.1. K-homology 2.2. Contact manifolds 2.3. Example: A hypoelliptic (but not elliptic) Fredholm operator 2.4. The problem 2.5. The solution 2.6. Characteristic class formula 2.7. Examples 3. Outline of the proof 4. Noncommutative topology of contact structures 4.1. The C * -algebra of the Heisenberg group 4.2. U (n) symmetry 4.3. Bargmann-Fok space 4.4. Quantization 4.5. Associated bundles 4.6. Inverting the Connes-Thom isomorphism 4.7. Noncommutative Poincaré duality 5.The index theorem as a commutative triangle 5.1. The Heisenberg filtration 5.2. Heisenberg pseudodifferential calculus 5.3. The Heisenberg symbol in K-theory 5.4. The "Choose an operator" maps 5.5. The hypoelliptic index theorem in K-homology 6. Computation of the K-cycle 6.1. Perturbing the symbol 6.2. Suspension in K-theory 6.3. The general formula 6.4. Toeplitz operators 6.5. Second order scalar operators References Paul Baum thanks Dartmouth College for the generous hospitality provided to him via the Edward Shapiro fund. Erik van Erp thanks Penn State University for a number of productive and enjoyable visits.

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis on the Heisenberg group.For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the K-theory of a noncommutative C * -algebra that is associated to the algebra of symbols. There is a canonical map from this analytic K-theory group to the ordinary cohomology of the manifold, which gives a de Rham class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators.Our methods derive from Connes' tangent groupoid proof of the index theorem.

We give an intrinsic (coordinate-free) construction of the tangent groupoid of a filtered manifold.Comment: 11pages. This paper was formerly part of the preprint arXiv:1511.01041 (version 3). This version involves a significant reorganization. It also adds a section on the adiabatic groupoid of a Lie groupoid with filtered Lie algebroi