noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
K-theory as a generalized homology theory.
In terms of KK-theory, the $K$-homology of a C*-algebra $A$ is $KK(A,\mathbb{C})$.
There are various useful ways to present K-homology classes.
See at Baum-Douglas geometric cycle.
Let $(X,g)$ be a Riemannian manifold. Let
be the algebra of continuous sections of the exterior bundle vanishing at infinity, and let
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
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
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
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:
An expository account of the analytic model:
Twisted K-homology:
Last revised on July 26, 2023 at 09:09:00. See the history of this page for a list of all contributions to it.