A spectral sequence for the homology of a finite algebraic delooping

TL;DR

This paper develops two spectral sequences to compute the homology of E_n-algebras, linking algebraic structures to topological loop space homology and providing concrete examples and decompositions.

Contribution

It introduces novel spectral sequences for E_n-algebra homology and connects these to Hochschild homology and derived functors, expanding computational tools.

Findings

Spectral sequences for E_n-algebra homology constructed.

Identification of Hodge decomposition summands with derived functors.

Homology computations for Hochschild cochains, polynomial algebras, and loop spaces.

Abstract

In the world of chain complexes E_n-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic E_n-homology of an E_n-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 K\"ahler differentials.