Analytic K-homology - Imported Knowlpedia Content

Definition

For a separable complex $C^*$ -algebra $A$ , analytic K-homology is the $\mathbb Z/2$ -graded abelian group

K^j(A)=KK^j(A,\mathbb C),\qquad j\in\{0,1\}.

Its degree-zero cycles are even Fredholm modules over $A$ , and its degree-one cycles are odd Fredholm modules. Classes are obtained by the stable-homotopy equivalence generated by unitary equivalence, operator homotopy, and addition or removal of degenerate cycles. Direct sum gives addition. A $*$ -homomorphism $A\to B$ pulls a $B$ -cycle back to an $A$ -cycle, so analytic K-homology is contravariant in the algebra.

Cycle picture and group structure

In the bounded picture, a cycle consists of a representation and an operator that is self-adjoint, involutive, and central modulo compact operators, with a grading in even degree. A degenerate cycle has all compact-defect expressions equal to zero and represents the zero class. Stabilization permits harmless degenerate summands, while operator homotopy records continuous deformation of the Fredholm data. Connes identifies $K_*(A)=KK(A,\mathbb C)$ with stable homotopy classes of Fredholm modules in Chapter IV, Section 1.

Bott periodicity supplies the two-periodic grading, so no further independent degrees are needed. The $KK$ -notation also places K-homology inside Kasparov's bivariant theory.

Relation to spaces and K-theory

If $X$ is compact Hausdorff, analytic K-homology of $C(X)$ agrees with the operator-theoretic model of topological K-homology of $X$ . For smooth compact manifolds, elliptic operators provide fundamental examples of its classes. Connes, Chapter IV, Section 1, discussion following Proposition 2.

Analytic K-homology is the homological partner of operator K-theory. The even index pairing and odd index pairing give

K_j(A)\times K^j(A)\longrightarrow\mathbb Z,

implemented by Fredholm indices of compressed operators.

Spectral triples and examples

A spectral triple is unbounded geometric data rather than, by itself, an equivalence class in the bounded cycle model. Its bounded transform produces a Fredholm module and hence an analytic K-homology class.

For $A=\mathbb C$ ,

K^0(\mathbb C)\cong\mathbb Z,\qquad K^1(\mathbb C)=0.

The even integer is detected by the ordinary Fredholm index. More generally, a Dirac operator on a closed spin manifold determines the manifold's fundamental analytic K-homology class.

Conventions and scope

Notation varies: some authors write $K_j(A)$ for $KK_j(A,\mathbb C)$ , while others reserve lower indices for K-theory and write $K^j(A)$ for K-homology. The core uses upper indices to keep the pairing $K_j(A)\times K^j(A)$ visually distinct.

Separability is the standard hypothesis in elementary $KK$ -theory treatments. Broader versions exist for nonseparable algebras, but their set-theoretic and categorical conventions should be stated explicitly.

References