nLab K-homology

Idea

K-theory as a generalized homology theory.

In terms of KK-theory, the $K$-homology of a C*-algebra $A$ is $KK(A,\mathbb{C})$.

Presentations

There are various useful ways to present K-homology classes.

By geometric cycles

See at Baum-Douglas geometric cycle.

By Fredholm modules and Dirac operators

Let $(X,g)$ be a Riemannian manifold. Let

$$C_\tau(X) \coloneqq \Gamma_0(\wedge^\bullet T^\ast X)$$

be the algebra of continuous sections of the exterior bundle vanishing at infinity, and let

$$\mathcal{H} \coloneqq L^2(\wedge^\bullet T^\ast X)$$

be the space of square integrable sections of the exterior bundle. Write $\mathcal{D} = d + d^\ast$ for the Kähler-Dirac operator and $\mathcal{F} = \mathcal{D} (1 + \mathcal{D}^2)^{-1/2}$. Then $(\mathcal{H}, \mathcal{F})$ is a Fredholm-Hilbert module which hence represents an element

$$[d_X + d^\ast_X] \in KK(C_\tau(X), \mathbb{C}) \,.$$

Now assume that $X$ carries a spin^c structure. Then there exists a vector bundle $S \to X$ such that $C_\tau(X) = End(S)$ and hence a Morita equivalence $C_\tau(X) \simeq_{Morita} C_0(X)$ with the algebra of continuous functions vanishing at infinity.

Let then

$$H \coloneqq L^2(S)$$

and write $D$ for the Spin^c Dirac operator and $F \coloneqq D (1+ D^2)^{-1/2}$.

Then under the above Morita equivalence these two Fredholm-Hilbert modules represent the same element in K-homology

$$[d_X + d^\ast_X] = [D] \in KK(C_0(X), \mathbb{C}) \,.$$

Related concepts

• K-cohomology

• Baum-Douglas geometric cycles

References

The “geometric model” of K-homology (cf. Baum-Douglas geometric cycle) is due to:

• Paul Baum, R. Douglas, K-homology and index theory, in: R. Kadison (ed.), Operator Algebras and Applications, Proceedings of Symposia in Pure Math. 38 AMS (1982) 117-173 [ams:pspum-38-1]

• Paul Baum, R. Douglas. Index theory, bordism, and K-homology, Contemp. Math. 10 (1982) 1-31

Analytic formulation via KK-theory:

• Gennady Kasparov, Equivariant KK-theory and the Novikov conjecture, Inventiones Mathematicae, 91 (1988) 147 [doi:10.1007/BF01404917, web]

An expository account of the analytic model:

• Nigel Higson, John Roe, Analytic K-homology, Oxford Mathematical Monographs. Oxford University Press, Oxford (2000) [ISBN: 9780198511762]

Twisted K-homology:

• Christopher Douglas, On the Twisted K-Homology of Simple Lie Groups, Topology 45 (2006), 955-988 (arXiv:math/0402082)