Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. 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 topological bivariant K-theory 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 K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

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 self-maps to an equivariant K-homology class. We compute the Lefschetz invariants for self-maps of finite-dimensional 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.     

The Baum-Connes assembly map, delocalization and the Chern character

For a discrete group Γ, we explicitly describe the rational Baum-Connes assembly map in “homological degree ?2” and show that in this domain it factors through the algebraic K-theory of the complex group ring of Γ. We also state and prove a delocalization property for , namely expressing it rationally in terms of the Novikov assembly map. Finally, we give a handicrafted construction of the delocalized equivariant Chern character (in the analytic language) and prove that it coincides with the equivariant Chern character of Lück (Invent. Math. 149 (2002) 123-152) (defined in the topological framework).     

Caractère de Chern bivariant

We construct a bivariant Chern character with values in Jones-Kassel's bivariant cyclic cohomology. This is done forK-theoretic objects such as idempotents, bimodules, quasi-homomorphisms à la Cuntz and extensions of algebras.

     

Bivariant Chern character and longitudinal index

In this paper we consider a family of Dirac-type operators on fibration P→B equivariant with respect to an action of an étale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map Φ in the cyclic cohomology. A particular case of this result is Connes' index theorem for étale groupoids [A. Connes, Noncommutative Geometry, Academic Press, 1994] in the case of fibrations.     

The relation between the Baum-Connes Conjecture and the Trace Conjecture

We prove a version of the L ²-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology 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 Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K ₀(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 [15]. Oblatum 10-IV-2001 & 18-X-2001?Published online: 15 April 2002     

On the commutativity of localized Chern characters with refined Gysin homomorphisms

We give a simple proof of the fact that the localized Chern characters of Baum, Fulton and MacPherson commute with the refined Gysin homomorphisms of [3]. This has been proved in [3] in the context of bivariant intersection theory using the technique of deformation to the normal cone. Our proof is more elementary in the sense that it avoids such a deformation and relies on a commutativity formula of these Chern characters with effective Cartier divisors. From this formula we also derive easily that the localized Chern characters pass to rational equivalence.     

Bivariant Chern classes for morphisms with nonsingular target varieties

W. Fulton and R. MacPherson posed the problem of unique existence of a bivariant Chern class—a Grothendieck transformation from the bivariant theory F of constructible functions to the bivariant homology theory H. J.-P. Brasselet proved the existence of a bivariant Chern class in the category of embeddable analytic varieties with cellular morphisms. In general however, the problem of uniqueness is still unresolved. In this paper we show that for morphisms having nonsingular target varieties there exists another bivariant theory of constructible functions and a unique bivariant Chern class γ: . Partially supported by Grant-in-Aid for Scientific Research (C) (No. 15540086+No. 17540088), the Japanese Ministry of Education, Science, Sports and Culture.     

On a generalised Connes-Hochschild-Kostant-Rosenberg theorem

The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalising some fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also identified here with the twisted Chern character.     

Equivariant Spectral Triples on the Quantum SU(2) Group

We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L ₂-space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SU_q(2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU(2), and we prove that for p < 4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dimension p acting on the L ₂-space.The first author would like to acknowledge support from the National Board of Higher Mathematics, India.     

Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert-Smith conjecture. The Hilbert-Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.     

On the Equivalence of Integral T^k-Cohomology Chern Numbers and T^k-K-Theoretic Chern Numbers

This paper mainly deals with the question of equivalence between equivariant cohomology Chern numbers and equivariant K-theoretic Chern numbers when the transformation group is a torus.By using the equivariant Riemann-Roch relation of AtiyahHirzebruch type,it is proved that the vanishing of equivariant cohomology Chern numbers is equivalent to the vanishing of equivariant K-theoretic Chern numbers.     

Chern character in equivariant entire cyclic cohomology

We construct the Chern character in the equivariant entire cyclic cohomology. We prove a general index theorem for theG-invariant Dirac operator.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065.     

On Grothendieck transformations in Fulton-MacPherson’s bivariant theory

W. Fulton and R. MacPherson have introduced a notion unifying both covariant and contravariant theories, which they called a Bivariant Theory. A transformation between two bivariant theories is called a Grothendieck transformation. The Grothendieck transformation induces natural transformations for covariant theories and contravariant theories. In this paper we show some general uniqueness and existence theorems on Grothendieck transformations associated to given natural transformations of covariant theories. Our guiding or typical model is MacPherson’s Chern class transformation c_∗:F→H_∗. The existence of a corresponding bivariant Chern class γ:F→H was conjectured by W. Fulton and R. MacPherson, and was proved by J.-P. Brasselet under certain conditions.     

A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof

We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K ₀-theory for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K ₀-theory induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart in [BaFQ]. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut (cf. [B2, Par. (l), p. 353] and also Ch. Soulé’s question in [SABK, 1.5, p. 162]). Oblatum 22-I-1999 & 20-II-2001?Published online: 4 May 2001     

Twisted equivariant K-theory with complex coefficients

Using a global version of the equivariant Chern character, wedescribe the complexified twisted equivariant K-theory of aspace with a compact Lie group action in terms of fixed-pointdata. We apply this to the case of a compact group acting onitself by conjugation and relate the result to the Verlindealgebra and to the Kac numerator at q=1. Verlinde's formulais also discussed in this context. Received February 28, 2007.     

Transgression on hyperkähler manifolds and generalized higher torsion forms

We propose a generalization of the Hodge dd_c-lemma to the case of hyperk?hler manifolds. As an application we derive a global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperk?hler manifolds. In Section 3 we consider the fourth order transgression for the infinite-dimensional bundle arising from local families of hyperk?hler manifolds. We propose a local construction of the fourth order transgression of the Chern character form. We derive an explicit expression for the arising hypertorsion differential form. Its zero-degree part may be expressed in terms of the Laplace operators defined on the fibres of the local family.