Sur l'algèbre de cohomologie cyclique d'un espace 1-connex. Applications à la géométrie des variétés. (On the algebra of cyclic cohomology of a 1-connected space. Applications to the geometry of manifolds) (Q1822380)

Let X be a 1-connected pointed space and k a characteristic zero field. It has been proved by \textit{T. G. Goodwillie} [Topology 24, 187-215 (1985; Zbl 0569.16021)] and \textit{D. Burghelea} and \textit{Z. Fiedorowicz} [ibid. 25, 303-317 (1986)] that cyclic cohomology \(HC^ *(X,k)\) is isomorphic to the cohomology of the space \(X^{S^ 1} \times _{S^ 1} ES^ 1.\) Furthermore, \(HC^ *(X,k)\) is a graded module over the polynomial ring \(H^ *(BS^ 1)=k[u]\) with \(\deg u=2.\) In this paper, the author shows that, for any 1-connected space X, we have a decomposition, as k[u]-graded modules: \(HC^ *(X)=k[u]\oplus V^ *\) where \(V^ *\) is a torsion module. Applications of this formula to Waldhausen algebraic K-theory are given. We have also results about rational homotopy of the group of diffeomorphisms of compact manifolds (precisely for \(CP^ n\), \(HP^ n\), \(U(n)/U(k)\times U(n-k)).\)