nLab Baum-Douglas geometric cycles



For a given manifold XX a Baum-Douglas geoemtric cycle on XX is data consisting of a submanifold QXQ \hookrightarrow X carrying a vector bundle EXE\to X such that this represents a class in K-homology under a suitable equivalence relation.

Viewed as a correspondence of the form

X(Q,E)* X \stackrel{}{\leftarrow} (Q,E) \to \ast

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

A generalization to geometric (co)-cocycles for KK-theory is in section 3 of

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.