Algebraic & Geometric Topology

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

Benoit Fresse

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715210

Digital Object Identifier
doi:10.2140/agt.2011.11.747

Mathematical Reviews number (MathSciNet)
MR2782544

Zentralblatt MATH identifier
1238.57034

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.agt/1513715210


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