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

Abstract

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.