On the homotopy and equivalence of locally \((G, \Lambda)\)-principal pseudogroups (Q689809)

Let \(G\) be a 1-connected Lie group and \(\Gamma\) a dense subgroup of \(G\). The locally \((G,\Gamma)\)-principal pseudogroups are closely related to foliations, mainly to transversally complete foliations. In a previous paper [C. R. Acad. Sci., Paris, Sér. I 314, No. 6, 463- 466 (1992; Zbl 0756.57018)] the author showed the existence of a space \(B_{G,\Gamma}\) such that there is a one to one correspondence between the set of equivalence classes of locally \((G,\Gamma)\)-principal pseudogroups with base a \(CW\)-complex \(W\) and the set of homotopy classes of continuous functions from \(W\) to \(B_{G,\Gamma}\). There the result asserting that if the base is paracompact then equivalence or homotopy classes of locally \((G,\Gamma)\)-principal pseudogroups are essentially the same was used. In this paper the author proves not only that result but also that if two differentiable locally \((G,\Gamma)\)-principal pseudogroups are homotopic then they are actually differentially equivalent.