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

Benoit Fresse

Algebraic & Geometric Topology 11 (2011) 747–838

We proved in a previous article that the bar complex of an E–algebra inherits a natural E–algebra structure. As a consequence, a well-defined iterated bar construction Bn(A) can be associated to any algebra over an E–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–algebras. We use this effective definition to prove that the n–fold bar construction admits an extension to categories of algebras over En–operads.

Then we prove that the n–fold bar complex determines the homology theory associated to the category of algebras over an En–operad. In the case n = , we obtain an isomorphism between the homology of an infinite bar construction and the usual Γ–homology with trivial coefficients.

iterated bar complex, $E_n$–operad, module over operad, homology theory
Mathematical Subject Classification 2010
Primary: 57T30
Secondary: 55P48, 18G55, 55P35
Received: 6 December 2010
Accepted: 17 December 2010
Published: 12 March 2011
Benoit Fresse
UFR de Mathématiques
Université Lille 1 - Sciences et Technologies
Cité Scientifique - Bâtiment M2
59655 Villeneuve d’Ascq Cedex