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A formula for characteristic classes of foliated \(\mathbb{R}^ n\)-products. (English) Zbl 0848.57027
Summary: For a topological group \(G\), \(G^\delta\) stands for the same group with discrete topology and \(\overline {G}\) denotes the homotopy theoretical fiber of the identity map \(G^\delta\to G\). \(\overline {G}\) is realized as a group given by the fiber product \(G^\delta \times_G PG= \{g,l)\in G^\delta \times PG: l(1)= g\}\), where \(PG\) is the path space of \(G\), based at the identity. The classifying space \(B\overline {G}\) classifies flat \(G\)-bundles (i.e., \(G^\delta\)-bundles) with trivialized \(G\)-bundle structures up to isomorphism. Such bundles are called foliated products. As is stated by D. B. Fuks [Usp. Mat. Nauk 28, No. 2(170), 3-17 (1973; Zbl 0272.57012)] and T. Tsuboi [Foliations, Proc. Symp., Tokyo 1983, Adv. Stud. Pure Math. 5, 37-120 (1985; Zbl 0674.57023)], \(B\overline {G}\) is obtained as a geometric realization of \(S^* (G) /G\) where \(S^* (G)\) is the semisimplicial complex of singular simplices with right \(G\)-action. Thus a typical \(k\)-simplex in \(B\overline {G}\) is represented by a map \(\sigma: \Delta^k\to G\), which we can assume smooth when \(G\) has a smooth manifold structure. Here, \(\Delta^k\) is a Euclidean \(k\)-simplex in \(\mathbb{R}^{k+1}\). In this paper, we deal with the case \(G= \text{Diff} \mathbb{R}^n\) and give a formula for a certain class of cocycles in \(B\text{ Diff } \mathbb{R}^n\) (see §4) which come from the secondary characteristic classes of foliations in terms of \(\sigma\). In a previous paper [A fête of topology, Pap. Dedic. Itiro Tamura, 49-62 (1988; Zbl 0848.57026), see the review above] we gave a formula for the Godbillon-Vey classes in \(\text{Diff } \mathbb{R}^n\). We generalize it to so-called residual classes in the secondary classes of foliations. The explicit formula is given in Theorem 1 and in Theorem 2. Throughout the paper, we work in the smooth category and all objects are assumed to be of class \(C^\infty\) unless otherwise specified.

57R30 Foliations in differential topology; geometric theory
58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
57R20 Characteristic classes and numbers in differential topology