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Index theory and NonCommutative Geometry II. Dirac Operators and Index Bundles
"... Abstract. When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K−theory index. This result gives a concrete connection between the topology of the foliation and the longit ..."
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Abstract. When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K−theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the NovikovShubin invariants of the operator is improved.
A NONHAUSDORFF ÉTALE GROUPOID
, 812
"... We present an example of a nonHausdorff, étale, essentially principal groupoid for which two results, known to hold in the Hausdorff case, fail. These results are: (A) the subalgebra of continuous functions on the unit space is maximal abelian, and (B) every nontrivial ideal of the reduced groupoid ..."
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We present an example of a nonHausdorff, étale, essentially principal groupoid for which two results, known to hold in the Hausdorff case, fail. These results are: (A) the subalgebra of continuous functions on the unit space is maximal abelian, and (B) every nontrivial ideal of the reduced groupoid C*algebra has a nontrivial intersection with the subalgebra of continuous functions on the unit space. 1. Introduction. This paper is concerned with étale groupoids [9,1,8,10,4]. A topological groupoid G is said to be étale if its unit space G (0) is locally compact and Hausdorff, and the range map “r ” (and consequently also the source map “s”) is a local homeomorphism. One may or may not assume the global topology of G to be Hausdorff but, while
THE TWISTED HIGHER HARMONIC SIGNATURE FOR FOLIATIONS
, 711
"... Abstract. We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation F of a compact Riemannian manifold M with coefficients in a leafwise U(p, q)flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted ..."
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Abstract. We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation F of a compact Riemannian manifold M with coefficients in a leafwise U(p, q)flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated. 1.
TDuality for Torus Bundles with HFluxes via Noncommutative Topology, II: the highdimensional case . . .
, 2008
"... ..."
Twodimensional measured laminations of positive Euler characteristic
, 1998
"... this paper could be written (and an earlier draft was written) entirely in terms of product decompositions, with no mention of branched surfaces. However, many concepts are more easily formulated in branched surface language, and many constructions are more easily visualized using branched surfaces. ..."
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this paper could be written (and an earlier draft was written) entirely in terms of product decompositions, with no mention of branched surfaces. However, many concepts are more easily formulated in branched surface language, and many constructions are more easily visualized using branched surfaces. Given q : ! (B; D) as above, we define the mass of relative to D to be MassD () =
The equivariant analytic index for proper groupoid actions
, 2007
"... The paper constructs the analytic index for an elliptic pseudodifferential family of Lmoperators invariant under the proper action of a ρ,δ continuous family groupoid on a Gcompact, C∞,0 Gspace. ..."
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The paper constructs the analytic index for an elliptic pseudodifferential family of Lmoperators invariant under the proper action of a ρ,δ continuous family groupoid on a Gcompact, C∞,0 Gspace.
Lie groupoids and Lie algebroids in physics and noncommutative geometry
 Journal of Geometry and Physics
, 2006
"... Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified descript ..."
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Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified description of internal and external symmetries. What is shared by noncommutative geometry and physics is the importance of Connes’s idea of associating a C ∗algebra C ∗ (Γ) to a Lie groupoid Γ: in noncommutative geometry C ∗ (Γ) replaces a given singular quotient space by an appropriate noncommutative space, whereas in physics it gives the algebra of observables of a quantum system whose symmetries are encoded by Γ. Moreover, Connes’s map Γ ↦ → C ∗ (Γ) has a classical analogue Γ ↦ → A ∗ (Γ) in symplectic geometry due to Weinstein, which defines the Poisson manifold of the corresponding classical system as the dual of the socalled Lie algebroid A(Γ) of the Lie groupoid Γ, an object generalizing both Lie algebras and tangent bundles. Only a handful of physicists appear to be familiar with Lie groupoids and Lie algebroids, whereas the latter are practically unknown even to mathematicians working
EQUIVARIANT EMBEDDING THEOREMS AND TOPOLOGICAL INDEX MAPS
, 2009
"... The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally nonsingular map is a map ..."
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The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally nonsingular map is a map together with such a factorisation. These factorisations are models for the topological index map. Under some assumptions concerning the existence of equivariant vector bundles, any smooth map admits a normal factorisation, and two such factorisations are unique up to a certain notion of equivalence. To prove this, we generalise the Mostow Embedding Theorem to spaces equipped with proper groupoid actions. We also discuss orientations of normally nonsingular maps with respect to a cohomology theory and show that oriented normally nonsingular maps induce wrongway maps on the chosen cohomology theory. For Koriented normally nonsingular maps, we also get a functor to Kasparov’s equivariant KKtheory. We interpret this functor as a topological index map.
Induction for Banach Algebras, Groupoids and KK ban
, 2008
"... Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C ∗coefficients. To show these results, the functoriality of Lafforgue’s KKtheory for ..."
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Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C ∗coefficients. To show these results, the functoriality of Lafforgue’s KKtheory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L 1algebras (with Banach algebra coefficients). Keywords: Locally compact groupoid, KK bantheory, Banach algebra, BaumConnes conjecture, Bost conjecture;
Dualities in equivariant Kasparov theory
"... Abstract. We study several duality isomorphisms between equivariant bivariant Ktheory groups, generalising Kasparov’s first and second Poincaré duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised selfmaps. The second duality ..."
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Abstract. We study several duality isomorphisms between equivariant bivariant Ktheory groups, generalising Kasparov’s first and second Poincaré duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised selfmaps. The second duality is related to the description of bivariant Kasparov theory for commutative C ∗algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities apply to a universal proper Gspace. This is a basic requirement for the dual Dirac method and allows us to describe the Baum–Connes assembly map via localisation of categories. Contents