The cobar construction as a Hopf algebra
The author starts with the tensor algebra of the desuspension of the reduced chain complex of a 1-reduced simplicial set \(X\). Hence 1-reduced means that in the associated graded object to \(X\) the subsets \(X_0\) and \(X_1\) consist of single points. The tensor algebra admits a differential \(d_\Omega\) and the homology coincides with that of the loop space of a geometric realization of \(X\). J. F. Adams gave a formula for the differential, and the present author shows that over \(\mathbb{Z}\) we have the additional, natural structure of a `homotopy Hopf algebra'. This consists of a coassociative diagonal map, and a derivation homotopy; explicit formulae are obtained for both. Using this data it is possible to construct a differential Lie algebra, the differential being defined on the free rational Lie algebra of the original desuspension. This time the homology coincides with the rational homotopy groups of the loops on the realization. Again there is additional `homotopy Hopf' structure. The interest of this work is that it provides an elegant alternative description of D. Quillen's isomorphism between the rational homotopy categories of 1-reduced simplicial sets and free differential Lie algebras, together with previously unachieved finiteness results. As the author himself remarks, given the complexity of the rational homotopy groups of an arbitrary space \(X\), one cannot expect the formulae for the derivations to be straightforward, but their explicit nature certainly leads to the improvements to the existing theory to which the reviewer has alluded.
scientific article; zbMATH DE number 1186149