Calcul différentiel sur la variété diff+[infini](S1) et calcul homologique du groupe diff+[infini](S1)

Abstract : Our goal is to study the group G of smooth diffeomorphisms of the circle. In this proceding, we investigate tow directions : first one is to prove again that G is a Fréchet space. The second one is to get some results about homological group of G with coefficients in the G-modules of the vector spaces of smooth differentiel forms on the circle or the real numbers. In the beginning, we define differentiability on Fréchet spaces. These spaces are complete for Mackey topology. The Fréchet spaces are convenient. Then we prove again that G is locally homeomorph to the Fréchet space of smooth applications from the circle and taking there values in the real numbers. The group G is then a Fréchet manifold. Our proofs are more easy than F. Sergaert and J. Milnor ones. Afterwards, we recapture some results of A. Haefliger about the O-th homology group of a subgroup G' of g with coefficients in the G'-module of smooth differential forms on the circle. The later results are motivating our homological calculus of the 1-th homology group of G with coefficient in the same G-module. This calculus is very dependant of the resolution of the equation f=hog-h, where f is a smooth real fonction on the circle and g is a diffeomorphism in G. We get unicity and existence conditions of the equation solutions. This fact has enable us how constructing and caracterizing elements of this1-th homology group and the 2-th homology group of G with trivial coefficients in real numbers. These tow homological groups are not trivial
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Mohamed Salah Bouallagui. Calcul différentiel sur la variété diff+[infini](S1) et calcul homologique du groupe diff+[infini](S1). Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1994. Français. ⟨NNT : 1994METZ067S⟩. ⟨tel-01776864⟩

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