Algebra Structures on the Comparison of the Reduced Bar Construction and the Reduced W-Construction

For a simplicial augmented algebra K, Eilenberg-Mac Lane constructed a chain map . They proved that g is a reduction (homology isomorphism) and conjectured that it is also the injection of a contraction (special homotopy equivalence). The contraction is followed at once by using homological perturba...

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Bibliographic Details
Published in:Communications in algebra 2009-10, Vol.37 (10), p.3643-3665
Main Authors: Álvarez, V., Armario, J. A., Frau, M. D., Real, P.
Format: Article
Language:English
Subjects:
Bar and W-constructions
Contraction
Homological perturbation lemma
Simplicial algebras
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Summary:For a simplicial augmented algebra K, Eilenberg-Mac Lane constructed a chain map . They proved that g is a reduction (homology isomorphism) and conjectured that it is also the injection of a contraction (special homotopy equivalence). The contraction is followed at once by using homological perturbation techniques. If K is commutative, Eilenberg-Mac Lane proved that g is a morphism of DGA-algebras. The present article is devoted to proving that f and φ satisfy certain multiplicative properties (weaker than g) and showing how they can be used for computing in an economical way the homology of twisted cartesian products of two Eilenberg-Mac Lane spaces.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927870902747662