Iterated bar complexes of E-infinity algebras and homology theories

作者：Fresse Benoit^*

来源：Algebraic and Geometric Topology, 2011, 11(2): 747-838.

DOI：10.2140/agt.2011.11.747

We proved in a previous article that the bar complex of an E(infinity)-algebra inherits a natural E(infinity)-algebra structure. As a consequence, a well-defined iterated bar construction B(n)(A) can be associated to any algebra over an E(infinity)-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(infinity)-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 = infinity, we obtain an isomorphism between the homology of an infinite bar construction and the usual Gamma-homology with trivial coefficients.