On the localization of topological invariants. (English) Zbl 0769.53015
This paper developes the approach of formal differential geometry to the topological invariants which can be localized (such as the characteristic classes of vector bundles and the characteristic classes of automorphisms of vector bundles). Many of the known secondary forms and the relations between them are obtained from one universal construction: The universal space is introduced; this is a differential manifold whose points are jets of the differential geometric objects or fields from which the topological invariant under consideration is obtained. On a universal space having a given nature there is associated a universal field of the same nature; from which the universal form is obtained. The components of the universal form (which are homogeneous) are universal secondary classes. Several identities concerning these classes are obtained. The following important cases are studied: 1) The field is the connection in a bundle over a manifold; the corresponding topological invariants are the Chern classes.
2) The fields are the connection in a bundle and the fibrewise automorphism of this bundle; the topological invariant is the Chern character of the corresponding element of \(K^ 1\).
3) The field is the Quillen superconnection on a \(\mathbb{Z}_ 2\)-graded vector bundle; the topological invariant is the Chern character of the bundle. As a secondary form for one problem may be a primary form for another problem, this phenomenon is finally discussed.
2) The fields are the connection in a bundle and the fibrewise automorphism of this bundle; the topological invariant is the Chern character of the corresponding element of \(K^ 1\).
3) The field is the Quillen superconnection on a \(\mathbb{Z}_ 2\)-graded vector bundle; the topological invariant is the Chern character of the bundle. As a secondary form for one problem may be a primary form for another problem, this phenomenon is finally discussed.
Reviewer: N.L.Youssef (Giza)
MSC:
| 53C05 | Connections (general theory) |
| 58A20 | Jets in global analysis |
| 57R20 | Characteristic classes and numbers in differential topology |
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