Hodge decomposition for higher order Hochschild homology

Let Gamma be the category of finite pointed sets and F be a functor from Ga mma to the category of vector spaces over a characteristic zero field. Loda y proved that one has the natural decomposition pi(n)F(S-1) congruent to +( n)(i=0)(F), n greater than or equal to 0. We show that for any d greater th an or equal to 1, there exists a similar decomposition for pi(n)F(S-d). Her e Sd is a simplicial model of the d-dimensional sphere. The striking point is, that the knowledge of the decomposition for pi(n)(S-1) (respectively pi (n)F(S-2)) completely determines the decomposition of pi(n)F(S-d) for any o dd (respectively even) d. These results can be applied to the cohomology of the mapping space X-Sd, where X is a d-connected space. Thus Hedge decompo sition of H*(X-S1)and H*(X-S2) determines all groups H*(X-Sd), d greater th an or equal to 1. (C) 2000 Editions scientifiques et medicales Elsevier SAS .