Hodge decomposition for higher order Hochschild homology

Abstract

Let Γ be the category of finite pointed sets and F be a functor from Γ to the category of vector spaces over a characteristic zero field. Loday proved that one has the natural decomposition π nF(S 1)≅⊕ i=0 nH n (i)(F), n≥0. We show that for any d≥1, there exists a similar decomposition for π n F( S d ). Here S d is a simplicial model of the d-dimensional sphere. The striking point is, that the knowledge of the decomposition for π n F( S 1) (respectively π n F( S 2)) completely determines the decomposition of π n F( S d ) for any odd (respectively even) d. These results can be applied to the cohomology of the mapping space X S d , where X is a d-connected space. Thus Hodge decomposition of H ∗(X S 1 ) and H ∗(X S 2 ) determines all groups H ∗(X S d ),d≥1 .