×

Homotopy properties of classifying spaces of foliations. (Homotopieeigenschaften von klassifizierenden Räumen von Blätterungen.) (German) Zbl 0838.57020

Berlin: FB Math. u. Inf., FU Berlin, 73 S. (1993).
In [Topology 23, 233-244 (1984; Zbl 0555.57011)] Takashi Tsuboi constructed a subgroup \(G\) of \(\text{Diff}^+ (S^1)\) such that the Godbillon-Vey invariant classified transversely foliated \(S^1\)-bundles over surfaces with \(G\)-structure. As a consequence of this construction he obtained a subgroup of \(H_3 (B \Gamma_1; \mathbb{Z})\), the third integral homology group of the classifying space of codimension 1 foliations, which is isomorphic to \(\mathbb{R}\). In his thesis the author extends this to the codimension 2 case, exhibiting a subgroup of \(H_5 (B \Gamma_2; \mathbb{Z})\) isomorphic to \(\mathbb{R}^2\) and detected by the two generators of the Gelfand-Fuks cohomology in dimension 5 of the Lie algebra of formal vector fields on \(\mathbb{R}^2\). The difficult part in obtaining this result is to prove that \(H_3 (\text{Diff}^+ (S^2); \mathbb{Z})\) contains a subgroup isomorphic to \(\mathbb{R}^2\). \(\text{Diff}^+ S^1\) and \(\text{Diff}^+ S^2\) are considered as discrete groups in these investigations.
Reviewer: E.Vogt (Berlin)

MSC:

57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology

Citations:

Zbl 0555.57011
PDFBibTeX XMLCite