A Canonical Enriched Adams-hilton Model for Simplicial Sets

For any 1-reduced simplicial set K we define a canonical, coassociative coproduct on ΩC(K), the cobar construction applied to the normalized, integral chains on K, such that any canonical quasi-isomorphism of chain algebras from ΩC(K) to the normalized, integral chains on GK, the loop group of K, is a coalgebra map up to strong homotopy. Our proof relies on the operadic description of the category of chain coalgebras and of strongly homotopy coalgebra maps given in [HPS]. Introduction Let X be a topological space. It is, in general, quite difficult to calculate the algebra structure of the loop space homology H∗ΩX directly from the (singular or cubical) chain complex C∗ΩX. An algorithm that associates to a space X a differential graded algebra whose homology is relatively easy to calculate and isomorphic as an algebra to H∗ΩX is therefore of great value. In 1955 [AH], Adams and Hilton invented such an algorithm for the class of simply-connected CW-complexes, which can be summarized as follows. Let X be a CW-complex such that X has exactly one 0-cell and no 1-cells, and such that every attaching map is based with respect to the unique 0-cell of X. There exists a morphism of differential graded algebras inducing an isomorphism on homology—a quasi-isomorphism— θX : (TV, d) ≃ −→ C∗ΩX, such that θX restricts to quasi-isomorphisms (TV≤n, d) ≃ −→ C∗ΩXn+1, where Xn+1 denotes the (n + 1)-skeleton of X, TV denotes the free (tensor) algebra on a free, 1991 Mathematics Subject Classification. Primary: 55P35 Secondary: 16W30, 18D50, 18G35, 55U10, 55U35, 57T05, 57T30.