nLab Baum-Douglas geometric cycles

Idea

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

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

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).

References

The general idea of geometric cycles for homology theories goes back to

• William Fulton, Robert MacPherson, Categorical framework for the study of singular spaces, Memoirs of the AMS, 243, 1981

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

• Alain Connes, Georges Skandalis, The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20,

  no. 6, 1139–1183 (1984) (pdf)

and for equivariant KK-theory in

• Heath Emerson, Ralf Meyer, Bivariant K-theory via correspondences, Adv. Math. 225 (2010), 2883-2919 (arXiv:0812.4949)

For further such generalizations see at bivariant cohomology theory.

Generalization to twisted K-homology is in

• Bai-Ling Wang, Geometric cycles, index theory and twisted K-homology (arXiv:0710.1625)