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Hopf algebras up to homotopy. (English) Zbl 0681.55006
For each simply-connected CW complex X and any subring R of $${\mathbb{Q}}$$, the Adams-Hilton’s construction yields an associative differential graded R-algebra (dga) A(X). In order for a dga (A,d) to equal A(X), for some X, it must receive a coproduct $$\psi: (A,d)\to (A,d)\otimes (A,d)$$ homotopically associative, homotopically cocommutative and with a homotopy counit. Such a triple $$(A,d,\psi)$$ is called a “Hopf algebra up to homotopy”.
The main result is the following: Let $$(A,d,\psi)$$ be an R-Hopf algebra up to homotopy. We suppose that R contains $$n^{-1}$$ for $$n<p$$ and that A is the tensor algebra generated by the range $$A_ r$$ through $$A_{rp- 1}$$, then there is an R-differential graded Lie algebra $$(L,\delta)$$ such that $$U(L,\delta)\approx (A,d)$$ as Hopf-algebras up to homotopy.
In particular, if X is an r-connected CW complex of dimension $$\leq rp$$ and if R contains $$n^{-1}$$ for $$n<p$$, then there exists an R- differential graded Lie algebra $$(L,\delta)$$ such that the Adams-Hilton model of X is isomorphic to $$U(L,\delta)$$. Moreover for every prime $$q\geq p$$, $$q^{th}$$ powers vanish in $$\tilde H^*(\Omega X;{\mathbb{Z}}_ q)$$, and $$H_*(\Omega X;{\mathbb{Z}}_ q)$$ is primitively generated (Wilkerson’s conjecture).
These results are very nice and striking, in particular because they enable us to use commutative models for the cochains algebra on X with coefficients in $${\mathbb{Z}}_ q$$ for $$q>>0$$.
Reviewer: Y.Felix

##### MSC:
 55P35 Loop spaces 55P62 Rational homotopy theory 57T30 Bar and cobar constructions 18G55 Nonabelian homotopical algebra (MSC2010) 17B70 Graded Lie (super)algebras
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