By V. Villani
A. Banyaga: at the team of diffeomorphisms protecting an actual symplectic.- G.A. Fredricks: a few comments on Cauchy-Riemann structures.- A. Haefliger: Differentiable Cohomology.- J.N. Mather: at the homology of Haefliger’s classifying space.- P. Michor: Manifolds of differentiable maps.- V. Poenaru: a few comments on low-dimensional topology and immersion theory.- F. Sergeraert: l. a. classe de cobordisme des feuilletages de Reeb de S? est nulle.- G. pockets: Invariant de Godbillon-Vey et diff?omorphismes commutants.
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Extra resources for Differential Topology (C.I.M.E. Summer Schools, 73)
Sample text
For instance, let 7M M 1 be the groupoid whoje space of units is the bundle PM of frames of order one (cf. I,8). The elements of FM are the germs 1 1 at points of PM of the prolongation of local diffeomorphisms of M to PM (cf. I,8). A ~M-cocycleis a rM-structure with a trivialization of its B T (or ~ BT~) is the homotopic fiber to the morphism V : rn Gln mapping y normal bundle. The classifying space of the map Brn + BGln associated on its derivative. + ... + dx A dx on R o 2n Let n : R ~ -+ ~R~~ +by the ~ natural projection (x , ,x2n) (xl ,x 1.
32] Kobayashi-Nomizu. Foundations of differential geometry I and I1 - Interscience Tracts in Pure and Applied Math. (1963). L. Koszul. Homologie et Cohomologie des algebres de Lie. Bull. Soc. Math. de France 78 1950, 65-127. L. Koszul. Homologie des formes differentielles dlordre supIrieur. Ann. Sc. Ec. Norm. Sup. 7 (1974) 139-153. V. Losik. Cohomology of the Lie algebra of vector fields with coefficients in a trivial unitary representation - Functional Analysis 6 (1972), 24-36. [361 S. MacLane.
M' To prove exactness, one uses an homotopy operator defined exactly by the same 'formula as in II,3, lemma 2. /3n)rM z s , C * ( a t 3 ), and we conclude from the M n n second step, as in II,3,that the inclusion BY I,8 n*(p induces an isomomorphism in cohomology. For M the theorem is proved in the same way, PKbnbeing replaced Remark 1 3ne can prove a similar theorem for any groupoid of germs of a transitive Lie pseudogroup. For instance the differentiable cohomology of - is isomorphic to rn the Gelfand-Fuchs cohomology of th'eLie algebra of 2n formal symplectic vector fields on R .