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More characteristic invariants of foliated bundles. (English) Zbl 0561.57016

Differential geometry, 14th Semest. S. Banach Int. Math. Cent., Warsaw 1979, Banach Cent. Publ. 12, 9-22 (1984).
Let \(B(P(M,G),E,H,\xi)\) be a quadruple consisting of a principal \(G\)-bundle \(P(M,G)\) over a manifold \(M\), \(E\) a flat partial connection on \(P(M,G)\), \(H\) a Lie subgroup of \(G\) and \(\xi\) the homotopy class of an \(H\)-reduction of \(P(M,G)\). In a previous paper [Diss. Math. 222, 67 p. (1984; Zbl 0538.57013)] making use of the Chern-Weil homomorphism, the author defined characteristic invariants for such quadruples. A further generalization is done in the present paper.
Namely, the author shows that there exists a well-defined characteristic homomorphism \[ \Phi (B): \oplus H(F^ a_ bW(G)_ H)\rightarrow \oplus H(I^ a_ F/I^ b_ F) \] \((\oplus\) is taken for integers \(0\leq a<b)\) where \(W(G)_ H\) is the \(H\)-basic Weil algebra of \(G\), \(F^ a_ bA=F^ aA/F^ bA\) for the filtration \(F^ kW(G)=\oplus_{b\geq k}{\mathfrak g}^*\otimes S^ b{\mathfrak g}^*\) and \(I^ k_ F\) is the differential ideal of \(A^*(M)\) (de Rham complex) generated by the forms which are locally written as \(\sum \phi^{i_ 1}\wedge\dotsm\wedge \phi^{i_ k}\wedge \psi_{i_ 1\dots i_ k}\) \((\phi^ i\)’s are 1-forms which are zero when restricted to each leaf of the projected foliation \(F\) of \(P(M,G))\). The author then proves that for any foliation \(F\), \(\{p_ i(F)\}\) and \(\{s_{2i-1}(F)\}\) generate the ring of characteristic classes. Here \(p_ i(F)\in H^{2i}(I^ i_ F)\) and \(s_{2j+1}(F)\in H^{4j+1}(A^*(M)/I_ F^{2j+1})\) are the Pontryagin classes and the secondary Pontryagin classes of \(F\) which he previously defined.
[For the entire collection see Zbl 0546.00020.]

MSC:

57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology