Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Hopf algebras up to homotopy


Author: David J. Anick
Journal: J. Amer. Math. Soc. 2 (1989), 417-453
MSC: Primary 16A24; Secondary 55P15
DOI: https://doi.org/10.1090/S0894-0347-1989-0991015-7
MathSciNet review: 991015
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (A,d)$ denote a free $ r$-reduced differential graded $ R$-algebra, where $ R$ is a commutative ring containing $ {n^{ - 1}}$ for $ 1 \leq n < p$. Suppose a ``diagonal'' $ \psi :A \to A \otimes A$ exists which satisfies the Hopf algebra axioms, including cocommutativity and coassociativity, up to homotopy. We show that $ (A,d)$ must equal $ U(L,\delta )$ for some free differential graded Lie algebra $ (L,\delta )$ if $ A$ is generated as an $ R$-algebra in dimensions below $ rp$. As a consequence, the rational singular chain complex on a topological monoid is seen to be the enveloping algebra of a Lie algebra. We also deduce, for an $ r$-connected CW complex $ X$ of dimension $ \leq rp$, that the Adams-Hilton model over $ R$ is an enveloping algebra and that $ p$th powers vanish in $ {\tilde H^ * }(\Omega X;{{\mathbf{Z}}_p})$.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, On the cobar construction, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 409-412. MR 0079266 (18:59c)
  • [2] J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1955), 305-330. MR 0077929 (17:1119b)
  • [3] D. J. Anick, A model of Adams-Hilton type for fiber squares, Illinois J. Math. 29 (1985), 463-502. MR 786733 (86h:55009)
  • [4] M. Aubry and J.-M. Lemaire, Homotopies d'algèbres de Lie et de leurs algèbres enveloppantes, Algebraic Topology-Rational Homotopy, Lecture Notes in Math., no. 1318, Springer-Verlag, Berlin and New York, 1988, 26-30. MR 952569 (89k:55018)
  • [5] H. J. Baues, Algebraic homotopy, Cambridge Stud. Adv. Math., no. 15, Cambridge Univ. Press, Cambridge, 1989. MR 985099 (90i:55016)
  • [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), 219-242. MR 0431172 (55:4174)
  • [8] F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. (2) 109 (1979), 121-168. MR 519355 (80e:55024)
  • [9] C. A. McGibbon and C. Wilkerson, Loop spaces of finite complexes at large primes, Proc. Amer. Math. Soc. 96 (1986), 698-702. MR 826505 (87h:55015)
  • [10] J. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 0174052 (30:4259)
  • [11] H. Munkholm, DGA algebras as a Quillen model category, J. Pure Appl. Algebra 13 (1978), 221-232. MR 509162 (80m:55018)
  • [12] D. G. Quillen, Homotopical algebra, Lecture Notes in Math., no. 43, Springer-Verlag, Berlin and New York, 1967. MR 0223432 (36:6480)
  • [13] -, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205-295. MR 0258031 (41:2678)
  • [14] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1978), 269-331. MR 0646078 (58:31119)
  • [15] D. Tanré, Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Math., no. 1025, Springer-Verlag, Berlin and New York, 1983. MR 764769 (86b:55010)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 16A24, 55P15

Retrieve articles in all journals with MSC: 16A24, 55P15


Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1989-0991015-7
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society