In the world of chain complexes En-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic En-homology of an En-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvili's Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kähler differentials.

Let ξbe an SO(n)-bundle over a simple connected manifold M with a spin structure Q → M. The string class is an obstruction to h1 the structure group LSpin(n) of the loop group bundle LQ → LM to the universal central extension of LSpain(n) by the circle. We prove that the string class vanishes if and only if 1/2 the first Pontrjagin clsss of values when M is a compact simply connected homogeneous space of rank one, a simpiy connected 4 dimensional manifold or a finite product space of those manifolds. This result is deduced by using the Eclesberg spectral sequence converging to the mod p cohomology of LM whose E2-term to the Hochschild homology of the mod p cohomology algebra of M. The key to the consideration is existence of a morphism of algebras, which is injective below degree 3, from an important graded commutator algebra into the Hochschild homology of a certain graded commutative algebra.

The ordinary string class is an obstruction to lift the structure group LSpin(n) of a loop group bundle LQ → LM to the universal central extension of LSpin(n) by the circle. The vanishing problem of the ordinary string class and generalized string classes are considered from the viewpoint of the ring structure of the cohomology H*(M; R).