By V. Villani

A. Banyaga: at the staff of diffeomorphisms retaining 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**

31 M In fact, such a vector bundle is given by a smooth linear r representation of the group G in the fiber V of E (cf. I,8), and E is k the bundle with fiber V associated to the principal bundle PM of frames of order k. ,Y P of composable elements of I' M' For a more precise definition, let NkJrM = lim N ~ J be ~ the J ~ inverse limits of the manifolds N ~~r of sequences of r jets of k k M composable elements of Let t : NkJrM+ M be the map associating to M' k such a sequence the target of the first element.

Definition R) is the cohomology of the s i q Z e c o q l e x associated to the double complex c;(rM;nM) which is a The differentiabte cohomology H$(rM; subcomplex o p cp(rM; GI. Recall that, in this double complex, the first differential 6 is defined by formula *) of 111, 3 and the second one d by the exterior differential of forms. The szme definition could be applied to groupoids coming from Lie - pseudogroups, like Fn, rQ, etc. For instance Ha(Tn; R) is the cohomology of Functoriality P 0 Let G be the group Diff M with the C -topology and C t the same group with the discrete topology.

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.