K-homology Groups Research Articles

K-theory and index pairings for C*-algebras generated by $q$-normal operators

The paper presents a detailed description of the K-theory and K-homology of C*-algebras generated by $q$-normal operators, including generators and index pairing. The C*-algebras generated by $q$-normal operators can be viewed as a $q$-deformation of the quantum complex plane. In this sense, we find deformations of the classical Bott projections describing complex line bundles over the 2-sphere, but there are also simpler generators for the $K_0$-groups, for instance, one dimensional Powers-Rieffel type projections and elementary projections belonging to the C*-algebra. The index pairing between these projections and generators of the even K-homology group is computed, and the result is used to express the $K_0$-classes of quantized line bundles of any winding number in terms of the other projections.

Multiplicative structures and the twisted Baum-Connes assembly map

Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in 2003 or by Tu, Xu and Laurent-Gengoux in 2004. These twisted geometric K-homology groups are the left-hand sides of the twisted geometric Baum-Connes assembly maps recently constructed by Carrillo Rouse and Wang (2016), and hence one can transfer the multiplicative structure via the Baum-Connes map to the twisted K-theory groups whenever these assembly maps are isomorphisms.

Stratified fiber bundles, Quinn homology and brane stability of hyperbolic orbifolds

We revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and K-homology. We propose a definition of Type II string theory on such backgrounds that further carry stratified systems of fiber bundles, which generalize the more conventional orbifold and symmetric string backgrounds, together with a classification of wrapped branes by a suitable generalized homology theory. For spaces stratified fibered over hyperbolic orbifolds we use the algebraic K-theory of their fundamental groups and Quinn homology to derive criteria for brane stability in terms of an Atiyah–Hirzebruch type spectral sequence with its lift to K-homology. Stable D-branes in this setting carry stratified charges which induce new additive structures on the corresponding K-homology groups. We extend these considerations to backgrounds which support H-flux, where we use K-groups of twisted group algebras of the fundamental groups to analyze stability of locally symmetric spaces with K-amenable isometry groups, and derive stability conditions for branes wrapping the fibers of an Eilenberg–MacLane spectrum functor.

Geometric Baum-Connes assembly map for twisted Differentiable Stacks

We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric twisted K-homology groups and the geometric assembly map are well defined for Lie groupoids is new, as it was only sketched by Connes in his book for general Lie groupoids without any restrictive hypothesis, in particular for non Hausdorff Lie groupoids. We also prove the Morita invariance of the assembly map, giving thus a precise meaning to the geometric assembly map for twisted differentiable stacks. We discuss the relation of the assembly map with the associated assembly map of the $S^1$-central extension. The relation with the analytic assembly map is treated, as well as some cases in which we have an isomorphism. One important tool is the twisted Thom isomorphism in the groupoid equivariant case which we establish in the appendix.

Spectral triples and the geometry of fractals

We construct spectral triples for the Sierpinski gasket as infinite sums of unbounded Fredholm modules associated with the holes in the gasket and investigate their properties. For each element in the K-homology group we find a representative induced by one of our spectral triples. Not all of these triples, however, will have the right geometric properties. If we want the metric induced by the spectral triple to give the geodesic distance, then we will have to include a certain minimal family of unbounded Fredholm modules. If we want the eigenvalues of the associated generalized Dirac operator to have the right summability properties, then we get limitations on the number of summands that can be included. If we want the Dixmier trace of the spectral triple to coincide with a multiple of the Hausdorff measure, then we must impose conditions on the distribution of the summands over the gasket. For the elements of a large subclass of the K-homology group, however, the representatives are induced by triples having the desired geometric properties. We finally show that the same techniques can be applied to the Sierpinski pyramid.

On the Poincaré isomorphism in K-theory on manifolds with edges

The aim of this paper is to construct the Poincare isomorphism in K-theory on manifolds with edges. We show that the Poincare isomorphism can naturally be constructed in the framework of noncommutative geometry. More precisely, to a manifold with edges we assign a noncommutative algebra and construct an isomorphism between the K-group of this algebra and the K-homology group of the manifold with edges viewed as a compact topological space.

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphism.

Uniform K-homology theory

We define a uniform version of analytic K-homology theory for separable, proper metric spaces. Furthermore, we define an index map from this theory into the K-theory of uniform Roe C ∗ -algebras, analogous to the coarse assembly map from analytic K-homology into the K-theory of Roe C ∗ -algebras. We show that our theory has a Mayer–Vietoris sequence. We prove that for a torsion-free countable discrete group Γ, the direct limit of the uniform K-homology of the Rips complexes of Γ, lim d → ∞ K * u ( P d Γ ) , is isomorphic to K * top ( Γ , ℓ ∞ Γ ) , the left-hand side of the Baum–Connes conjecture with coefficients in ℓ ∞ Γ . In particular, this provides a computation of the uniform K-homology groups for some torsion-free groups. As an application of uniform K-homology, we prove a criterion for amenability in terms of vanishing of a “fundamental class”, in spirit of similar criteria in uniformly finite homology and K-theory of uniform Roe algebras.

Characters of Fredholm Modules and a Problem of Connes

A large part of research in noncommutative geometry is devoted to various far-reaching generalizations of the Atiyah–Singer index theory of elliptic operators on compact manifolds. Following Kasparov [K1], the notion of an elliptic operator on a manifold is replaced by that of a Fredholm module (or K-cycle) over an algebra of operators on Hilbert space. A number of index theorems for Fredholm modules associated to geometric data have been obtained by Kasparov [K1], Connes (see [C3] and the literature cited therein) and others. In order to handle concrete index problems it is not only necessary to dispose of an index theorem but also to provide an index formula which allows the explicit calculation of indices. The classical index formula of Atiyah and Singer is obtained from the index theorem by applying the Chern character in topological K-homology. This motivates the search for character formulas of Fredholm modules (K-cycles) that define a Chern character on the K-homology groups of an algebra of operators. It was this search for a Chern character in K-homology which led A. Connes to the invention of cyclic cohomology [C2]. He obtained various explicit character formulas [C1,2,3], which depend on the degree of analytic regularity (summability) of the given Fredholm module. Whereas the classical index problems of Atiyah–Singer are all finitely summable in the sense of [C2] there are many examples of Fredholm modules over (noncommutative) algebras which are infinite dimensional (not finitely summable). Characters of Fredholm modules have been calculated in many finitely summable cases, but as far as the author knows the character of a θ-summable, infinitedimensional (unbounded) Fredholm module [C1] has not yet been determined in a single case.

Homotopy classification of elliptic operators on stratified manifolds

We give a homotopy classification of elliptic operators on a stratified manifold. Namely, we establish an isomorphism between the set of elliptic operators modulo stable homotopy and the K-homology group of the manifold. By way of application, we obtain an explicit formula for the obstruction of Atiyah-Bott type to the existence of Fredholm problems in the case of stratified manifolds.

Homotopy classification of elliptic operators on stratified manifolds

We find the stable homotopy classification of elliptic operators on stratified manifolds. Namely, we establish an isomorphism of the set of elliptic operators modulo stable homotopy and the K-homology group of the singular manifold. As a corollary, we obtain an explicit formula for the obstruction of Atiyah–Bott type to making interior elliptic operators Fredholm.

Asymptotic morphisms, K-homology and Dirac operators

Using asymptotic morphisms between graded C*-algebras, we construct for every open m-dimensional spin manifold M a fundamental class in the m-th analytic K-homology group of M. This class is associated to the not necessarily essentially self-adjoint Dirac operator on M. A careful treatment is given of the main properties of these fundamental K-homology classes.

K -homology of C * -categories and symmetric spectra representing K -homology

We define K-homology groups K * (𝒞) for small C * -categories 𝒞 in terms of Hilbert modules over the C * -category 𝒞. We also define a functor A f from the category of small C * -categories into the category of C * -algebras and show that there is a natural isomorphism $K_*(\cal C)\cong K_*(A^f_{\cal C})$ . In addition, we give an easy construction of a functor ${\mathbb K}$ from the category of C * -algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor ${\mathbb K}$ comes with a natural isomorphism $K_*(A)\cong \pi_*({\mathbb K}(A))$ for C * -algebras A. It then follows that the composition ${\mathbb K}$ ○A f provides a functor that can be used in the Davis-Luck approach for constructing the Baum-Connes map.

K-theory and S-duality: starting over from square 3

Recently Maldacena, Moore, and Seiberg (MMS) have proposed a physical interpretation of the Atiyah-Hirzebruch spectral sequence, which roughly computes the K-homology groups that classify D-branes. We note that in IIB string theory, this approach can be generalized to include NS charged objects and conjecture an S-duality covariant, nonlinear extension of the spectral sequence. We then compute the contribution of the MMS double-instanton configuration to the derivation d_5. We conclude with an M-theoretic generalization reminiscent of 11-dimensional E_8 gauge theory.

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 L2-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 SUq(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 L2-space.

Triplets spectraux pour les variétés à singularité conique isolée

We use elliptic operators of Fuchs type on an even dimensional pseudomanifold with an isolated singularity to construct spectral triples. This class of operators includes Dirac operators with coefficients in flat bundles in the radial direction and, under some hypothesis, these operators generate the even K-homology group tensorised by ℂ of the pseudomanifold. Moreover, their Chern character is computed.

Triplets spectraux pour les pseudovariétés à singularité conique isolée

We use elliptic operators of Fuchs type on an even-dimensional pseudomanifold with an isolated singularity to construct spectral triples. This class of operators includes Dirac operators with coefficients in flat bundles in the radial direction and, under some hypothesis, these operators generate the even K-homology group tensorized by ¢, of the pseudomanifold. Moreover, their Chern character is computed.

Analytic K-theory on manifolds with corners

Let Y be a compact manifold. In [ 1 ] Atiyah proposed an analytic delinition of the K-homology groups K,(Y), defined abstractly as the dual theory to the Atiyah-Hirzebruch K(cohomology)-theory. These ideas were developed by Kasparov [15] and Brown, Douglas, and Fillmore [lo]. Cycles for this analytic K-homology theory are “abstract elliptic operators,” i.e., bounded linear operators between Hilbert spaces satisfying additional conditions derived from the properties of elliptic (pseudo-) differential operators. In particular elliptic pseudodifferential operators between sections of vector bundles define such cycles. The homology class of the cycle depends only on the principal symbol and indeed only on the K-cohomology class it represents on the cotangent bundle. In fact it had already been shown, in essence, by Atiyah that the resulting map