Iterated bar complexes of E-infinity algebras and homology theories

Details Iterated bar complexes of E-infinity algebras and homology theories Iterated bar complexes of E-infinity algebras and homology theories

2008-10-28

arxiv.org/abs/0810.5147v7

Mathematics

Algebraic Topology

71 pages. Concluding section reduced and appendix of the former version removed.

Scientific paper

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 complex B^n(A) 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 a category of algebras over E_n-operads. For n infinite, we obtain an isomorphism between the homology of an infinite bar construction and the usual Gamma-homology with trivial coefficients.

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