A spectral sequence for the homology of a finite algebraic delooping

Abstract

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.2000 Mathematics Subject Classification. Primary 16E40, 18G40; Secondary 55P35.

E n -spectra and Dyer-Lashof operations

“…The group-completion theorem allows us to relate the homology of a delooping to certain nonabelian derived functors [69]. Similar spectral sequences computing E n -homology of chain complexes have been studied by Richter and Ziegenhagen [79].…”

Cellular $E_k$-algebras

“…We will not develop those formulae, as we have-as yet-no need for them. We will not make use of this spectral sequence, but it has been studied in some detail by Richter-Ziegenhagen [RZ14], especially in even characteristic.…”