Algebraic & Geometric Topology

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

Benoit Fresse

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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.

Article information

Algebr. Geom. Topol., Volume 11, Number 2 (2011), 747-838.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57T30: Bar and cobar constructions [See also 18G55, 55Uxx]
Secondary: 55P48: Loop space machines, operads [See also 18D50] 18G55: Homotopical algebra 55P35: Loop spaces

iterated bar complex $E_n$–operad module over operad homology theory


Fresse, Benoit. Iterated bar complexes of $E$–infinity algebras and homology theories. Algebr. Geom. Topol. 11 (2011), no. 2, 747--838. doi:10.2140/agt.2011.11.747.

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