group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A Baum-Douglas geometric cycle on a given manifold $X$ is a representative for K-homology classes on $X$. It is given by a submanifold $Q \hookrightarrow X$ equipped with spin^c structure and with a complex vector bundle. The equivalence relation identifying such data that represents the same K-homology class includes a compatible bordism 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).
In string theory Baum-Douglas cycles constitute one formalization of the concept of D-brane carrying a Chan-Paton gauge field (Reis-Szabo 05, Szabo 08): the submanifold $Q$ represents the worldvolume of the D-brane and the complex vector bundle it carries the Chan-Paton gauge field.
The original articles 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.
A generalization to twisted homology is discussed in section of
A generalization to geometric (co)-cocycles for KK-theory is in section 3 of
no. 6, 1139–1183 (1984) (pdf)
More generally, a construction of general homology theories in a similar fashion is discussed in
and a construction of bivariant cohomology theories in this spirit is in
The interpretation as a formalization of D-branes in string theory is highlighted in
Rui Reis, Richard Szabo, Geometric K-Homology of Flat D-Branes ,Commun.Math.Phys. 266 (2006) 71-122 (arXiv:hep-th/0507043)
Richard Szabo, D-branes and bivariant K-theory, Noncommutative Geometry and Physics 3 1 (2013): 131. (arXiv:0809.3029)
Last revised on December 28, 2016 at 12:53:53. See the history of this page for a list of all contributions to it.