Differential Topology (C.I.M.E. Summer Schools, 73) by V. Villani

By V. Villani

A. Banyaga: at the workforce of diffeomorphisms maintaining 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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Example 2. The groupoid I' (or M rn). aD 'I M is the groupoid of germs of C -diffeomorphisms of the differentiable manifold M, with the sheaf topology. An element y of I'M is the germ at x e M of a diffeomorphism g of a nbhd U of x on some open set of M. A basis for the fopen set in rM is obtained by taking the g e m ' of g at the points of U. The space of units is the manifold H. When M is R ~ ,then TM will be noted rn. is an open covering of a manifold X, a rM-cocycle over % If- such that the yii : U.

A continuous homomotphi'sm h : r' induces a morphism of complexes h* : C*(r;A) if y : I+%-+ r . + -+ of topological groupoids C*(r'. ; h* 4). Passing to cohomology, and composing with the natural map v 2)+H(X ; c)(cf. 223), we get the characteristic homomor- phism s*(c*(r ;A)) - E*(x ;A Y ~ Example of a non trivial cocycle Let Example 2,i). p-fo- on ~ ~ ( c f . Let a c C1(l' log I y-l(x) '1 ,n 0) be the cochain associating to y the function -1 where (y ) '(x) is the jacobian of y at x. = t(y). This "-7 is a 1-cocycle whose cohomology class is not trivial.

Thi5orie des faisceaux, Hermann Paris (1964). [21] A. Grotlendieck. Sur quelques points d'algsbre homologique. Tohoku Math. Journ. 9 (1957), 119-183. 1221 V. Guillemin-S. Sternberg. Deformation Theory of Pseudogroup structures, Memoires AMS No 64 (1966). [231 V. Cohomology of vector fields on Manifolds, Advance in Math 10, 192-220 (1973). [241 A. Haefliger. Homotopy and Integrability - Lecture Notes in Math. [2i] A. Haefliger. Sur les classes caractgristiques des feuilletages - SCminaire Bourbaki annee 1971-1972, No 412.

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