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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

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