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27
The relation between the BaumConnes conjecture and the trace conjecture
 Invent. Math
"... We prove a version of the L 2index Theorem of Atiyah, which uses the universal centervalued trace instead of the standard trace. We construct for Gequivariant Khomology an equivariant Chern character, which is an isomorphism and lives over the ring Z ⊂ Λ G ⊂ Q obtained from the integers by inver ..."
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Cited by 17 (10 self)
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We prove a version of the L 2index Theorem of Atiyah, which uses the universal centervalued trace instead of the standard trace. We construct for Gequivariant Khomology an equivariant Chern character, which is an isomorphism and lives over the ring Z ⊂ Λ G ⊂ Q obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the BaumConnes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K0(C ∗ r (G)) → R takes values in Λ G. The original Trace Conjecture predicted that its image lies in the additive subgroup of R generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [13].
Index theory and NonCommutative Geometry I. Higher Families Index Theory
, 2005
"... We prove an index theorem for foliated manifolds. We do so by constructing a push forward map in cohomology for a Koriented map from an arbitrary manifold to the space of leaves of an oriented foliation, and by constructing a ChernConnes character from the Ktheory of the compactly supported smoo ..."
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Cited by 12 (7 self)
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We prove an index theorem for foliated manifolds. We do so by constructing a push forward map in cohomology for a Koriented map from an arbitrary manifold to the space of leaves of an oriented foliation, and by constructing a ChernConnes character from the Ktheory of the compactly supported smooth functions on the holonomy groupoid of the foliation to the Haefliger cohomology of the foliation. Combining these with the ConnesSkandalis topological index map and the classical Chern character gives a commutative diagram from which the index theorem follows immediately.
Bivariant Ktheory via correspondences
, 2008
"... Abstract. We use correspondences to define a purely topological equivariant bivariant Ktheory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth Koriented maps by a class of Koriented normal maps, which are ..."
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Cited by 9 (2 self)
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Abstract. We use correspondences to define a purely topological equivariant bivariant Ktheory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth Koriented maps by a class of Koriented normal maps, which are maps together with a certain factorisation. Our construction does not use any special features of equivariant Ktheory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories. We formulate necessary and sufficient conditions for certain duality isomorphisms in the geometric bivariant Ktheory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant Ktheory to Ktheory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, both bivariant Ktheories agree if there is such a duality isomorphism. 1.
Quantisation commutes with reduction at discrete series representations of semisimple groups
, 2008
"... Using the analytic assembly map that appears in the BaumConnes conjecture in noncommutative geometry, we generalise the Guillemin– Sternberg conjecture that ‘quantisation commutes with reduction ’ to (discrete series representations of) semisimple groups G with maximal compact subgroups K acting co ..."
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Cited by 7 (5 self)
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Using the analytic assembly map that appears in the BaumConnes conjecture in noncommutative geometry, we generalise the Guillemin– Sternberg conjecture that ‘quantisation commutes with reduction ’ to (discrete series representations of) semisimple groups G with maximal compact subgroups K acting cocompactly on symplectic manifolds. We prove this generalised statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements g ∗ se, the set of elements of g ∗ with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that M = G ×K N, for a compact Hamiltonian Kmanifold N. The proof comes down to a reduction to the compact case. This reduction is based on a ‘quantisation commutes with induction’principle, and involves a notion of induction of Hamiltonian group actions. This principle, in turn, is based on a version of the naturality of the assembly map for the inclusion K ֒ → G.
Analytic Novikov for topologists
 Proceedings of the 1993 Oberwolfach Conference on the Novikov Conjecture, volume 226 of LMS Lecture Notes
, 1995
"... Abstract. We explain for topologists the “dictionary ” for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgerytheoretic proofs. In particular, we try to explain the following points: (1) Why do the analytic proofs of the Novikov conjecture require the int ..."
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Cited by 6 (1 self)
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Abstract. We explain for topologists the “dictionary ” for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgerytheoretic proofs. In particular, we try to explain the following points: (1) Why do the analytic proofs of the Novikov conjecture require the introduction of C ∗algebras? (2) Why do the analytic proofs of the Novikov conjecture all use Ktheory instead of Ltheory? Aren’t they computing the wrong thing? (3) How can one show that the index map µ or β studied by operator theorists matches up with the assembly map in surgery theory? (4) Where does “bounded surgery theory ” appear in the analytic proofs? Can one find a correspondence between the sorts of arguments used by analysts and the controlled surgery arguments used by topologists? The literature on the Novikov conjecture (see [FRR]) consists of several different kinds of papers. Most of these fall into two classes: those based on topological arguments, usually involving surgery theory, and those based on analytic arguments, usually involving index theory. The purpose of this note is to “explain ” the second class of papers to those familiar with the first class. I do not intend here to give a detailed sketch of the Kasparov KKapproach to the Novikov conjecture (for which the key details appear in [Kas4], [Fac2], and [KS]), since this has already been done in the convenient expository references [Fac1], [Kas2], [Kas3], [Bla], and [Kas5]. Nor do I intend to explain the approach to the Novikov conjecture taken by
THE STRONG NOVIKOV CONJECTURE FOR LOW DEGREE COHOMOLOGY
, 705
"... ABSTRACT. We show that for each discrete group Γ, the rational assembly map K∗(BΓ) ⊗ Q → K∗(C ∗ maxΓ) ⊗ Q is injective on classes dual to Λ ∗ ⊂ H ∗ (BΓ; Q), where Λ ∗ is the subring generated by cohomology classes of degree at most 2 (and where the pairing uses the Chern character). Our result im ..."
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Cited by 5 (2 self)
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ABSTRACT. We show that for each discrete group Γ, the rational assembly map K∗(BΓ) ⊗ Q → K∗(C ∗ maxΓ) ⊗ Q is injective on classes dual to Λ ∗ ⊂ H ∗ (BΓ; Q), where Λ ∗ is the subring generated by cohomology classes of degree at most 2 (and where the pairing uses the Chern character). Our result implies homotopy invariance of higher signatures associated to classes in Λ ∗. This consequence was first established by ConnesGromovMoscovici [4] and Mathai [9]. Our approach is based on the construction of flat twisting bundles out of sequences of almost flat bundles as first described in our work [5]. In contrast to the argument in [9], our approach is independent of (and indeed gives a new proof of) the result of HilsumSkandalis [6] on the homotopy invariance of the index of the signature operator twisted with bundles of small curvature. 1.
Equivariant Lefschetz maps for simplicial complexes and smooth manifolds
 Math. Ann
"... Abstract. Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for selfmaps to an equivariant Khomol ..."
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Abstract. Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for selfmaps to an equivariant Khomology class. We compute the Lefschetz invariants for selfmaps of finitedimensional simplicial complexes and smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in this case. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Lück and Rosenberg. 1.
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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Cited by 3 (1 self)
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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.