Modular cohomology class from the viewpoint of characteristic class. (English) Zbl 0647.46055
Geometric methods in operator algebras, Proc. US-Jap. Semin., Kyoto/Jap. 1983, Pitman Res. Notes Math. Ser. 123, 375-386 (1986).
[For the entire collection see Zbl 0632.00012.]
This paper is an attempt to link the secondary characteristic classes in foliation theory to operator theory. Let F be a \(C^{\infty}\)-foliation on the manifold M and \(\delta\) a module of transverse measure on the holonomy groupoid of F, corresponding to a nowhere vanishing \(C^{\infty}\)-density of M. Let \(h_ 1(F,L)\) denote the first leaf invariant with respect to a leaf L of F, then it is proved that the modular cohomology class \(\iota\) [Log \(\delta\) ] of F has property that \(i^*\circ \iota [Log \delta]=-2\pi h_ 1(F,L).\)
This paper is an attempt to link the secondary characteristic classes in foliation theory to operator theory. Let F be a \(C^{\infty}\)-foliation on the manifold M and \(\delta\) a module of transverse measure on the holonomy groupoid of F, corresponding to a nowhere vanishing \(C^{\infty}\)-density of M. Let \(h_ 1(F,L)\) denote the first leaf invariant with respect to a leaf L of F, then it is proved that the modular cohomology class \(\iota\) [Log \(\delta\) ] of F has property that \(i^*\circ \iota [Log \delta]=-2\pi h_ 1(F,L).\)
Reviewer: Hou Jinchuan
MSC:
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 57R30 | Foliations in differential topology; geometric theory |
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
