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Contribution of noncommutative geometry to index theory on singular manifolds
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This survey of the work of the author with several collaborators presents the way groupoids appear and can be used in index theory. We define the general tools, and apply them to the case of manifolds with corners, ending with a topological index theorem.
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... The groupoid T nc X, called "The noncommutative tangent space of X", was introduced independently in [28] and in [12] where it was proven to be the K-Poincaré dual of the conic pseudomanifold associated to X. ...
We give a cohomological formula for the index of a fully elliptic
pseudodifferential operator on a manifold with boundary. As in the classic case
of Atiyah-Singer, we use an embedding into an euclidean space to express the
index as the integral of a cohomology class depending in this case on a
noncommutative symbol, the integral being over a -manifold called the
singular normal bundle associated to the embedding. The formula is based on a
K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is
drawn from Connes' tangent groupoid approach.
... For the topology, it is very easy to see that all the groupoid structures are compatible with the glueings we considered. We are going to consider a deformation groupoid T G X ([9]). This will be a natural generalisation of the Connes tangent groupoid of a smooth manifold, to the case with boundary. ...
In his book (II.5), Connes gives a proof of the Atiyah-Singer index theorem for closed manifolds by using deformation groupoids and appropiate actions of these on R^N. Following these ideas, we prove an index theorem for manifolds with boundary. Comment: 6 pages. Preprint submitted to the Academie des Sciences
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincare-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A* of a Lie algebroid.
In Part I of this paper (6) we proved various index theorems for manifolds with boundary including an extension of the Hirzebruch signature theorem. We now propose to investigate the geometric and topological implications of these theorems in a variety of contexts.
1. Introduction . The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula:
where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X . In particular if, near the boundary, X is isometric to the product Y x R ⁺ , the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H ² ( X , R) by an integral formula
where p 1 is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p 1 = (2π) ⁻² Tr R ² . It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general
We compute the K-theory, groups of the C*-algebra of the groupoid of a manifold with corners, in which the analytic index takes its values.
