Iterated bar complexes of E–infinity algebras and homology theories

Fresse, Benoit

doi:10.2140/agt.2011.11.747

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

Benoit Fresse

Algebraic & Geometric Topology 11 (2011) 747–838

DOI: 10.2140/agt.2011.11.747

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 $B^{n} (A)$ 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 $Γ$ –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

References

Publication

Received: 6 December 2010

Accepted: 17 December 2010

Published: 12 March 2011

Authors

Benoit Fresse
UMR CNRS 8524 UFR de Mathématiques Université Lille 1 - Sciences et Technologies Cité Scientifique - Bâtiment M2 59655 Villeneuve d’Ascq Cedex France
http://math.univ-lille1.fr/~fresse/

Volume 11, issue 2 (2011)