For a given manifold $X$ a Baum-Douglas geoemtric cycle on $X$ is data consisting of a submanifold $Q \hookrightarrow X$ carrying a vector bundle $E\to X$ such that this represents a class in K-homology under a suitable equivalence relation.
Viewed as a correspondence of the form
a Baum-Douglas geometric cycle is a special case of the spans that represent classes in KK-theory (between manifolds) according to (Connes-Skandalis 84, section 3).
The general idea of geometric cycles for homology theories goes back to
The original articles by Baum and Douglas are
Paul Baum, R. Douglas, K-homology and index theory: Operator Algebras and Applications (R. Kadison editor), volume 38 of Proceedings of Symposia in Pure Math., 117-173,
Providence RI, 1982. AMS.
Paul Baum, R. Douglas. Index theory, bordism, and K-homology, Contemp. Math. 10: 1-31 1982.
The proof that these geometric cycles indeed model all of K-homology is due to
Paul Baum, Nigel Higson, Thomas Schick, On the Equivalence of Geometric and Analytic K-Homology (arXiv:math/0701484)
Jeff Raven, An equivariant bivariant Chern character, PhD Thesis
A generalization to geometric (co)-cocycles for KK-theory is in section 3 of
no. 6, 1139–1183 (1984) (pdf)
and for equivariant KK-theory in
For further such generalizations see at bivariant cohomology theory.
Generalization to twisted K-homology is in
Last revised on August 14, 2013 at 19:24:47. See the history of this page for a list of all contributions to it.