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 -homology of a C*-algebra is .
There are various useful ways to present K-homology classes.
See at Baum-Douglas geometric cycle.
Let 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 for the Kähler-Dirac operator and . Then is a Fredholm-Hilbert module which hence represents an element
Now assume that carries a spin^c structure. Then there exists a vector bundle such that and hence a Morita equivalence with the algebra of continuous functions vanishing at infinity.
Let then
and write for the Spin^c Dirac operator and .
Then under the above Morita equivalence these two Fredholm-Hilbert modules represent the same element in K-homology
Gennady Kasparov, Equivariant KK-theory and the Novikov conjecture, Inventiones Mathematicae, vol. 91, p.147 (web)
Christopher Douglas, On the Twisted K-Homology of Simple Lie Groups, Topology 45 (2006), 955-988 (arXiv:math/0402082)
Last revised on October 20, 2019 at 05:31:04. See the history of this page for a list of all contributions to it.