## Hopf algebras up to homotopy

HTML articles powered by AMS MathViewer

- by David J. Anick
- J. Amer. Math. Soc.
**2**(1989), 417-453 - DOI: https://doi.org/10.1090/S0894-0347-1989-0991015-7
- PDF | Request permission

## 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 $p\text {th}$ powers vanish in ${\tilde H^ * }(\Omega X;{{\mathbf {Z}}_p})$.## References

- J. F. Adams,
*On the cobar construction*, Proc. Nat. Acad. Sci. U.S.A.**42**(1956), 409–412. MR**79266**, DOI 10.1073/pnas.42.7.409 - J. F. Adams and P. J. Hilton,
*On the chain algebra of a loop space*, Comment. Math. Helv.**30**(1956), 305–330. MR**77929**, DOI 10.1007/BF02564350 - David J. Anick,
*A model of Adams-Hilton type for fiber squares*, Illinois J. Math.**29**(1985), no. 3, 463–502. MR**786733** - 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). MR**952569**, DOI 10.1007/BFb0077792 - Hans Joachim Baues,
*Algebraic homotopy*, Cambridge Studies in Advanced Mathematics, vol. 15, Cambridge University Press, Cambridge, 1989. MR**985099**, DOI 10.1017/CBO9780511662522
H. J. Baues, S. Halperin, and J.-M. Lemaire, - H. J. Baues and J.-M. Lemaire,
*Minimal models in homotopy theory*, Math. Ann.**225**(1977), no. 3, 219–242. MR**431172**, DOI 10.1007/BF01425239 - F. R. Cohen, J. C. Moore, and J. A. Neisendorfer,
*Torsion in homotopy groups*, Ann. of Math. (2)**109**(1979), no. 1, 121–168. MR**519355**, DOI 10.2307/1971269 - 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. MR**826505**, DOI 10.1090/S0002-9939-1986-0826505-X - John W. Milnor and John C. Moore,
*On the structure of Hopf algebras*, Ann. of Math. (2)**81**(1965), 211–264. MR**174052**, DOI 10.2307/1970615 - Hans J. Munkholm,
*DGA algebras as a Quillen model category. Relations to shm maps*, J. Pure Appl. Algebra**13**(1978), no. 3, 221–232. MR**509162**, DOI 10.1016/0022-4049(78)90009-9 - Daniel G. Quillen,
*Homotopical algebra*, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR**0223432** - Daniel Quillen,
*Rational homotopy theory*, Ann. of Math. (2)**90**(1969), 205–295. MR**258031**, DOI 10.2307/1970725 - Dennis Sullivan,
*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 269–331 (1978). MR**646078** - Daniel Tanré,
*Homotopie rationnelle: modèles de Chen, Quillen, Sullivan*, Lecture Notes in Mathematics, vol. 1025, Springer-Verlag, Berlin, 1983 (French). MR**764769**, DOI 10.1007/BFb0071482

*The uniqueness of rational homotopy*(in preparation).

## Bibliographic Information

- © Copyright 1989 American Mathematical Society
- 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