K-homology (changes) in nLab

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Contents

Idea

K-theory as a generalized homology theory.

In terms of KK-theory, the KK-homology of a C*-algebra AA is KK(A,ℂ)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)(X,g) be a Riemannian manifold. Let

C τ(X)≔Γ 0(∧ •T *X) 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

ℋ≔L 2(∧ •T *X) \mathcal{H} \coloneqq L^2(\wedge^\bullet T^\ast X)

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

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

Now assume that XX carries a spin^c structure. Then there exists a vector bundle S→XS \to X such that C τ(X)=End(S)C_\tau(X) = End(S) and hence a Morita equivalence C τ(X)≃ MoritaC 0(X)C_\tau(X) \simeq_{Morita} C_0(X) with the algebra of continuous functions vanishing at infinity.

Let then

H≔L 2(S) H \coloneqq L^2(S)

and write DD for the Spin^c Dirac operator and F≔D(1+D 2) −1/2F \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 X *]=[D]∈KK(C 0(X),ℂ). [d_X + d^\ast_X] = [D] \in KK(C_0(X), \mathbb{C}) \,.

• K-cohomology

• Baum-Douglas geometric cycles

References

• Gennady Kasparov, Equivariant KK-theory and the Novikov conjecture, Inventiones Mathematicae, vol. 91, p.147 (web)

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](https://bookstore.ams.org/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](https://doi.org/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, Oxford Science (2000) Publications. [[ISBN: ISBN: 9780198511762](https://global.oup.com/academic/product/analytic-k-homology-9780198511762?cc=nl&lang=en&)] 9780198511762

Twisted K-homology:

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