A generalization of manifold and its characteristic classes
The notion of an \(R_ n\)-set has been introduced by the author in Funkts. Anal. Prilozh. 21, No.3, 38-52 (1987; Zbl 0632.57016). The structures of \(R_ n\)-sets may be defined on the sets of transitivity classes of local diffeomorphisms on n-manifolds and on the sets of leaves of foliations of codimension n. The characteristic classes of \(R_ n\)- sets are obtained by using the cohomologies \(H^*(W_ n)\), \(H^*(W_ n,GL_ n)\), \(H^*(W_ n,O(n))\), where \(W_ n\) is the Lie algebra of formal vector fields on \({\mathbb{R}}^ n\), \(GL_ n=GL(n,R)\), \(O(n)=the\) orthogonal group. The Pontryagin classes and the classes induced by the characteristic classes of foliations may be found among these characteristic classes. The Pontryagin classes are real but are no longer integer valued. The introduced characteristic classes are not trivial in the case of \(H^*(W_ 1,GL_ 1)\).
transitivity classes of local diffeomorphisms on n-manifolds
scientific article; zbMATH DE number 4176016