K-homology Class Research Articles

A transverse index theorem in the calculus of filtered manifolds

We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland, and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols, and show a Poincaré duality type result linking the class of an operator and its symbol.

Levinson's theorem as an index pairing

We build on work of Kellendonk, Richard, Tiedra de Aldecoa and others to show that the wave operators for Schrödinger scattering theory on Rn generically have a particular form. As a consequence, Levinson's theorem can be interpreted as the pairing of the K-theory class of the unitary scattering operator and the K-homology class of the generator of the group of dilations.

Equivariant spectral triple for the quantum group Uq(2) for complex deformation parameters

Let q=|q|eiπθ be a nonzero complex number such that |q|≠1 and consider the compact quantum group Uq(2). For θ∉Q∖{0,1}, we obtain the K-theory of the underlying C⁎-algebra C(Uq(2)). We construct a spectral triple on Uq(2) which is equivariant under its own comultiplication action. The spectral triple obtained here is even, 4+-summable, non-degenerate, and the Dirac operator acts on two copies of the L2-space of Uq(2). The K-homology class of the associated Fredholm module is shown to be nontrivial.

Locally equivalent quasifree states and index theory

We consider quasifree ground states of Araki’s self-dual canonical anti-commutation relation algebra from the viewpoint of index theory and symmetry protected topological (SPT) phases. We first review how Clifford module indices characterise a topological obstruction to connect pairs of symmetric gapped ground states. This construction is then generalised to give invariants in with A a -algebra of allowed deformations. When A = C*(X), the Roe algebra of a coarse space X, and we restrict to gapped ground states that are locally equivalent with respect X, a K-homology class is also constructed. The coarse assembly map relates these two classes and clarifies the relevance of K-homology to free-fermionic SPT phases.

Transversals, duality, and irrational rotation

An early result of Noncommutative Geometry was Connes’ observation in the 1980’s that the Dirac-Dolbeault cycle for the 2 2 -torus T 2 \mathbb {T}^2 , which induces a Poincaré self-duality for T 2 \mathbb {T}^2 , can be ‘quantized’ to give a spectral triple and a K-homology class in K K 0 ( A θ ⊗ A θ , C ) \mathrm {KK}_0(A_\theta \otimes A_\theta , \mathbb {C}) providing the co-unit for a Poincaré self-duality for the irrational rotation algebra A θ A_\theta for any θ ∈ R ∖ Q \theta \in \mathbb {R}\setminus \mathbb {Q} . Connes’ proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-zero integer b b , a finitely generated projective module L b \mathcal {L}_{b} over A θ ⊗ A θ A_\theta \otimes A_\theta by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope θ \theta and θ + b \theta + b , using the fact that these flows are transverse to each other. We then compute Connes’ dual of [ L b ] [\mathcal {L}_{b}] and prove that we obtain an invertible τ b ∈ K K 0 ( A θ , A θ ) \tau _{b}\in \mathrm {KK}_0(A_\theta , A_\theta ) , represented by an equivariant bundle of Dirac-Schrödinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such ‘ b b -twists’ and this allows us to describe the unit of Connes’ duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit – a kind of ‘quantized Thom class’ for the diagonal embedding of the noncommutative torus.

The index of $G$-transversally elliptic families. I

We define and study the index map for families of G -transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah–Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology class of an elliptic operator on the base. Our approach is based on the functorial properties of the intersection product, and relies on some constructions due to Connes–Skandalis and to Hilsum–Skandalis.

Elliptic operators and K-homology

If a differential operator D on a smooth Hermitian vector bundle S over a compact manifold M is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If D is also elliptic, then the Hilbert space of square integrable sections of S with the canonical left C(M)-action and the operator χ(D) for χ a normalizing function is a Fredholm module, and its K-homology class is independent of χ. In this expository article, we provide a detailed proof of this fact following the outline in the book “Analytic K-homology” by Higson and Roe.

A Dolbeault-Dirac Spectral Triple for Quantum Projective Space

The notion of a Kahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kahler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes' axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kahler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.

A Dolbeault–Hilbert complex for a variety with isolated singular points

Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum-Fulton-MacPherson.

On Analytic Todd Classes of Singular Varieties

AbstractLet $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ with $\pi :M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth }_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial }+\overline{\partial }^t$ on $M$. In the 2nd part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini–Study metric. First, assuming $\dim (V)\leq 2$, we compare the Baum–Fulton–MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial }$ complex. We show that there is no $L^2$-$\overline{\partial }$ complex on $(\operatorname{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum–Fulton–MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

K-Homology classes of elliptic uniform pseudodifferential operators

We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.

K-homological finiteness and hyperbolic groups

Abstract Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C * \mathrm{C}^{*} -algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C * \mathrm{C}^{*} -algebras associated to hyperbolic groups – the C * \mathrm{C}^{*} -crossed product for the boundary action, and the reduced group C * \mathrm{C}^{*} -algebra – have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.

Pseudo-Riemannian spectral triples and the harmonic oscillator

We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a K-homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classical pseudo-Riemannian manifolds.

K-homology class of the Dirac operator on a compact quantum group

By a result of Nagy, the C^*-algebra of continuous functions on the q -deformation G_q of a simply connected semisimple compact Lie group G is KK-equivalent to C(G) . We show that under this equivalence the K-homology class of the Dirac operator on G_q , which we constructed in an earlier paper, corresponds to that of the classical Dirac operator. Along the way we prove that for an appropriate choice of isomorphisms between completions of U_q\mathfrak g and U\mathfrak g a family of Drinfeld twists relating the deformed and classical coproducts can be chosen to be continuous in q .

Uniform version of Weyl–von Neumann theorem

We prove a "quantified" version of the Weyl-von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in the Voiculescu's theorem applied to commutative algebras. This allows considerable simplifications in uniform K-homology theory, namely it shows that one can represent all the uniform K-homology classes on a fixed Hilbert space with a fixed *-representation of C_0(X), for a large class of spaces X.

The equivariant index theorem in entire cyclic cohomology

AbstractLet G be a locally compact group acting smoothly and properly by isometries on a complete Riemannian manifold M, with compact quotient G\M. There is an assembly map which associates to any G-equivariant K-homology class on M, an element of the topological K-theory of a suitable Banach completion of the convolution algebra of continuous compactly supported functions on G. The aim of this paper is to calculate the composition of the assembly map with the Chern character in entire cyclic homology . We prove an index theorem reducing this computation to a cup-product in bivariant entire cyclic cohomology. As a consequence we obtain an explicit localization formula which includes, as particular cases, the equivariant Atiyah-Segal-Singer index theorem when G is compact, and the Connes-Moscovici index theorem for G-coverings when G is discrete. The proof is based on the bivariant Chern character introduced in previous papers.

Torus Equivariant Spectral Triples for Odd-Dimensional Quantum Spheres Coming from C *-Extensions

The torus group (S1)l+1 has a canonical action on the odd-dimensional sphere Sq2l+1. We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial K-homology class thus generalizing our earlier results for SUq(2). We also relate the triple we construct with the C*-extension \({0\longrightarrow \mathcal{K}\otimes C(S^1)\longrightarrow C(S_q^{2\ell+3}) \longrightarrow C(S_q^{2\ell+1}) \longrightarrow 0.}\)

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

We construct a quantum version of the SU(2) Hopf bundle S 7→S 4. The quantum sphere S 7 q arises from the symplectic group Sp q (2) and a quantum 4-sphere S 4 q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S 4 q ) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S 4. We compute the fundamental K-homology class of S 4 q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU q (2) on S 7 q such that the algebra A(S 7 q ) is a non-trivial quantum principal bundle over A(S 4 q ) with structure quantum group A(SU q (2)).

Unbounded symmetric operators in K-homology and the Baum–Connes conjecture

Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an “addition formula” for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the K-homology of its classifying space and the analytic K-theory of its C * -algebra, in close connection with the Baum–Connes assembly map. For groups classified by a 2-complex, this allows to reformulate the Baum–Connes conjecture.

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.