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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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Cited by 8 (7 self)
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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.
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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Cited by 6 (3 self)
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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.
Compactly supported analytic indices for Lie groupoids”, preprint arXiv:0803.2060
, 2008
"... Abstract. For any Lie groupoid we construct an analytic index morphism taking values in a modified K −theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over ..."
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Cited by 3 (1 self)
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Abstract. For any Lie groupoid we construct an analytic index morphism taking values in a modified K −theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows in particular to prove a more primitive version of the ConnesSkandalis Longitudinal index Theorem for foliations, that is, an index theorem taking values in a group which pairs with Cyclic cocycles. As other application, for D a G −PDO elliptic operator with associated index ind D ∈ K0(C ∞ c (G)), we prove that the pairing < ind D, τ>, with τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)] ∈ K 0 (A ∗ G). The result is completely general for Étale groupoids. We discuss some potential applications to the Novikov’s conjecture.
Twisted longitudinal index theorem for foliations and wrong way functoriality
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THE HIGHER FIXED POINT THEOREM FOR FOLIATIONS I. HOLONOMY INVARIANT CURRENTS
, 2010
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Supported by the Austrian Federal Ministry of Education, Science and Culture
, 2009
"... Abstract. The goal of this note is to outline a proof that the JLO bivariant cocycle associated with a family of Dirac type operators over a smooth fibration, is entire for C ℓ+1 − C ℓ topologies, for any ℓ ≥ 0. This result is false if one uses the C ∞ topologies, or when the base manifold is infini ..."
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Abstract. The goal of this note is to outline a proof that the JLO bivariant cocycle associated with a family of Dirac type operators over a smooth fibration, is entire for C ℓ+1 − C ℓ topologies, for any ℓ ≥ 0. This result is false if one uses the C ∞ topologies, or when the base manifold is infinite dimensional. AMS classification: 58J30 Keywords: JLO, Noncommutative geometry, Families Index. 1. Preliminaries and notations In this note we define a bivariant JLO cocycle in terms of which we can reformulate the local families index theorem [6, 5]. We prove that our bivariant cocycle is entire in the sense of the formalism introduced by Meyer [10]. Thus we shall consider a fibration F → M π → B of closed manifolds endowed with smooth metrics. As can be seen from the proofs of the present paper, the main result remains true in larger categories like manifolds with corners [9] or Heisenberg manifolds [12], but we will not give the details here. The dimension of the fibers is denoted by p and the dimension of the base is p ′. We assume for simplicity that the fibers of our fibration are odd dimensional, so p is odd. The formulae in the even case are similar and are left as an exercise. We also fix a hermitian vector bundle E → M whose fibres are modules over the Clifford algebra
THE HIGHER HARMONIC SIGNATURE FOR FOLIATIONS
, 2008
"... Abstract. In this paper, we prove that the higher harmonic signature of a 4k dimensional oriented Riemannian foliation F of a compact Riemannian manifold M is a leafwise homotopy invariant. Contents ..."
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Abstract. In this paper, we prove that the higher harmonic signature of a 4k dimensional oriented Riemannian foliation F of a compact Riemannian manifold M is a leafwise homotopy invariant. Contents
Index theory and . . .
, 2009
"... This paper gives a survey of the index theory of tangentially elliptic and transversally elliptic operators on foliated manifolds as well as of related notions and results in noncommutative geometry. ..."
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This paper gives a survey of the index theory of tangentially elliptic and transversally elliptic operators on foliated manifolds as well as of related notions and results in noncommutative geometry.
A KTHEORETIC L 2INDEX THEOREM FOR FAMILIES
, 807
"... Abstract. We prove a Ktheoretic L 2index theorem for families of elliptic operators which in the case of a single operator reduces to the refined L 2index theorem of Lück from [14]. The proof employs embeddings into acyclic groups, following the main idea of [7]. ..."
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Abstract. We prove a Ktheoretic L 2index theorem for families of elliptic operators which in the case of a single operator reduces to the refined L 2index theorem of Lück from [14]. The proof employs embeddings into acyclic groups, following the main idea of [7].