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2011 Iterated bar complexes of $E$–infinity algebras and homology theories
Benoit Fresse
Algebr. Geom. Topol. 11(2): 747-838 (2011). DOI: 10.2140/agt.2011.11.747


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.


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Benoit Fresse. "Iterated bar complexes of $E$–infinity algebras and homology theories." Algebr. Geom. Topol. 11 (2) 747 - 838, 2011.


Received: 6 December 2010; Accepted: 17 December 2010; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1238.57034
MathSciNet: MR2782544
Digital Object Identifier: 10.2140/agt.2011.11.747

Primary: 57T30
Secondary: 18G55 , 55P35 , 55P48

Keywords: $E_n$–operad , homology theory , iterated bar complex , module over operad

Rights: Copyright © 2011 Mathematical Sciences Publishers


Vol.11 • No. 2 • 2011
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