The cobar construction as a Hopf algebra

Abstract

We show that the integral cobar construction ΩC * X of a 1-reduced simplicial set X has the structure of a homotopy Hopf algebra. This leads to a rational differential Lie algebra ℒ(X) given by a differential d ℒ on the free Lie algebra L(s −1 * X). The universal enveloping algebra Uℒ(X) is naturally isomorphic to ΩC * X⊗ℚ. Let X be a 1-reduced simplicial set and let s −1 * X be the desuspension of the reduced chain complex * X of X. The cobar construction is given by a differential d Ω on the tensor algebra T(s −1 * X) defined over ℤ. The homology of ΩC * X coincides with the homology of the loop space Ω|X|. A formula for d Ω was given by Adams [Ad]. We show that the cobar construction ΩC * X has over ℤ the additional natural structure of a “homotopy Hopf algebra” consisting of a coassociative diagonal and a derivation homotopy E:?ψ≃ψ. Explicit formulas for ψ and E are given in (2.9). The result is based on the geometric cobar construction in [B]. Using the fundamental theory of Anick [A] we derive from the data (ΩC * X,ψ,E) a differential Lie algebra which is defined by a differential d ℒ on the free rational Lie algebra L(s −1 * X). The homology of ℒ(X) coincides with the rational homotopy groups of Ω|X|. Moreover we describe a cocommutative diagonal together with a derivation homotopy H:ψ⊗ℚ≃φ and a natural iso- morphism of differential Hopf algebras Formulas for φ, H and β are obtained by an induction. A map f:X→Y between 1-reduced simplicial sets induces the map f *=L(s −1 * f): L(s −1 * X)→L(s −1 * Y) between free Lie algebras which is compatible with the differential d ℒ. Hence X↦ℒ(X) yields a canonical functor from 1-reduced simplicial sets to free differential Lie algebras. This functor induces an isomorphism of rational homotopy categories. In fact, ℒ is up to homotopy naturally equivalent to the functor λ=NPG of Quillen [Q] which is the composite of four intricate functors. We describe in [B”] inductively a sequence of elements β(2),β(3),… which determine the differential d ℒ and we compute the quadratic part of d ℒ. A complete formula for d ℒ, however, is highly intricate since it can be considered as a kind of “simplicial form” of the classical Hausdorff formula; see (4.9) (4). We cannot expect a simple form of the differential d ℒ since the homology of ℒ(X) yields the rational homotopy groups of X; that is This supplements the result of Kan [K] who gave a combinatorial description of π*(X) in terms of the free simplicial group GX. Hence π*(X) requires computations in free groups while the formula for π*(X)⊗ℚ requires only the computation of the homology of a chain complex of rational vector spaces. We point out that ℒ(X) is a chain complex of finite type if X has only finitely many non degenerate simplices. Hence in this case d ℒ yields a finite formula for π n (X)⊗ℚ. Such a formula was not yet achieved in the literature. For example using the Sullivan method one obtains π*(X)⊗ℚ only implicitly by computing a minimal model of the De Rham algebra A(X); see Bousfield-Gugenheim [BG]. Here A(X) is defined explicitly in terms of X but is not of finite type if X has only finitely many non degenerate simplices so that A(X) does not give a finite formula for π n (X)⊗ℚ. In this case the minimal model M X of A(X) is of finite type and hence also gives a finite formula for π n (X)⊗ℚ. There is, however, no explicit formula for M X in terms of X in the literature and M X is not natural in X. On the other hand also the model λ(X) of Quillen is considerably larger than ℒ(X) above and is only defined as the normalization of a simplicial object; see (4.9) (4) below.