Hopf algebras up to homotopy. (English) Zbl 0681.55006

Anick, David J.

J. Am. Math. Soc. 2, No. 3, 417-453 (1989).

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

Keywords:

universal enveloping algebra; homotopy commutative coalgebra; homology of loop spaces; simply-connected CW complex X; Adams-Hilton’s construction; associative differential graded R-algebra; Hopf algebra up to homotopy; differential graded Lie algebra; Adams-Hilton model

References:

[1]	J. F. Adams, On the cobar construction, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 409 – 412. · Zbl 0071.16404
[2]	J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305 – 330. · Zbl 0071.16403 · doi:10.1007/BF02564350
[3]	David J. Anick, A model of Adams-Hilton type for fiber squares, Illinois J. Math. 29 (1985), no. 3, 463 – 502. · Zbl 0546.55025
[4]	Marc Aubry and Jean-Michel Lemaire, Homotopies d’algèbres de Lie et de leurs algèbres enveloppantes, Algebraic topology — rational homotopy (Louvain-la-Neuve, 1986) Lecture Notes in Math., vol. 1318, Springer, Berlin, 1988, pp. 26 – 30 (French). · Zbl 0682.55008 · doi:10.1007/BFb0077792
[5]	Hans Joachim Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics, vol. 15, Cambridge University Press, Cambridge, 1989. · Zbl 0688.55001
[6]	H. J. Baues, S. Halperin, and J.-M. Lemaire, The uniqueness of rational homotopy (in preparation).
[7]	H. J. Baues and J.-M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977), no. 3, 219 – 242. · Zbl 0322.55019 · doi:10.1007/BF01425239
[8]	F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. (2) 109 (1979), no. 1, 121 – 168. · Zbl 0405.55018 · doi:10.2307/1971269
[9]	C. A. McGibbon and C. W. Wilkerson, Loop spaces of finite complexes at large primes, Proc. Amer. Math. Soc. 96 (1986), no. 4, 698 – 702. · Zbl 0594.55006
[10]	John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211 – 264. · Zbl 0163.28202 · doi:10.2307/1970615
[11]	Hans J. Munkholm, DGA algebras as a Quillen model category. Relations to shm maps, J. Pure Appl. Algebra 13 (1978), no. 3, 221 – 232. · Zbl 0409.55018 · doi:10.1016/0022-4049(78)90009-9
[12]	Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. · Zbl 0168.20903
[13]	Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205 – 295. · Zbl 0191.53702 · doi:10.2307/1970725
[14]	Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269 – 331 (1978). · Zbl 0374.57002
[15]	Daniel Tanré, Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics, vol. 1025, Springer-Verlag, Berlin, 1983 (French). · Zbl 0539.55001