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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.

##### MSC:
 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