#### Volume 11, issue 2 (2011)

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Iterated bar complexes of $E$–infinity algebras and homology theories

### Benoit Fresse

Algebraic & Geometric Topology 11 (2011) 747–838
##### Abstract

We proved in a previous article that the bar complex of an ${E}_{\infty }$–algebra inherits a natural ${E}_{\infty }$–algebra structure. As a consequence, a well-defined iterated bar construction ${\mathtt{B}}^{n}\left(A\right)$ can be associated to any algebra over an ${E}_{\infty }$–operad. In the case of a commutative algebra $A$, our iterated bar construction reduces to the standard iterated bar complex of $A$.

The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of ${E}_{\infty }$–algebras. We use this effective definition to prove that the $n$–fold bar construction admits an extension to categories of algebras over ${E}_{n}$–operads.

Then we prove that the $n$–fold bar complex determines the homology theory associated to the category of algebras over an ${E}_{n}$–operad. In the case $n=\infty$, we obtain an isomorphism between the homology of an infinite bar construction and the usual $\Gamma$–homology with trivial coefficients.

##### Keywords
iterated bar complex, $E_n$–operad, module over operad, homology theory
##### Mathematical Subject Classification 2010
Primary: 57T30
Secondary: 55P48, 18G55, 55P35