Their properties are the same as those of the corresponding classes in algebraic geometry: respect of the Adams operations, behavior with respect to products in K-theory and Deligne cohomology, and c_{1, 1} is the determinant map. Thus c_{2, 2} is known on Steinberg's symbols as pull-backs of the Heisenberg bundle and its connection. In the same manner, analytic Beilinson-Deligne Chern classes can be constructed (and the same properties hold).

In a second part, we specialize our study to c_{2, 2} (and smooth classes).
We show that, for a large class of manifolds (including Hausdorff,
finite-dimensional ones), it makes sense to take infinitesimal deformation
of c_{2, 2}. Such deformation yields a central extension of Lie algebras,
a cocycle of which is given by:

Then, this extension is a quotient of the Bloch's universal central extension of the C-Lie algebra sl C^\infty (X).

When X = S^1, this is the Segal's cocycle: the extension is the universal
central extension of Fréchet-Lie algebra sl C^\infty (X). This
shows that the infinitesimal deformation of c_{2, 2} identifies with the
infinitesimal deformation of the second Connes-Karoubi's regulator

This work is part of my thesis. The original (french) text (and other papers) are accessible by ftp or on my web page.

The dvi file uses the Xy-pic fonts, as well as the jknappen/dc fonts.

- 0304.bib (266 bytes)
- Regulateurs_Cinfty.dvi (120364 bytes) [September 29, 1998]
- Regulateurs_Cinfty.dvi.gz (46199 bytes)
- Regulateurs_Cinfty.pdf (145814 bytes)
- Regulateurs_Cinfty.ps (286177 bytes)
- Regulateurs_Cinfty.ps.gz (140797 bytes)

Daniel Delabre <delabre@picard.ups-tlse.fr>