Twisting cochains and duality between minimal algebras and minimal Lie algebras
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- by Richard M. Hain PDF
- Trans. Amer. Math. Soc. 277 (1983), 397-411 Request permission
Abstract:
An algebraic duality theory is developed between $1$-connected minimal cochain algebras of finite type and connected minimal chain Lie algebras of finite type by means of twisting cochains. The duality theory gives a concrete relationship between Sullivanâs minimal models, Chenâs power series connections and the various Lie algebra models of a $1$-connected topological space defined by Quillen, Allday, Baues-Lemaire and Neisendorfer. It can be used to compute the Lie algebra model of a space from the algebra model of the space and vice versa.References
- Christopher Allday, Rational Whitehead products and a spectral sequence of Quillen. II, Houston J. Math. 3 (1977), no. 3, 301â308. MR 474288
- H. J. Baues and J.-M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977), no. 3, 219â242. MR 431172, DOI 10.1007/BF01425239
- Edgar H. Brown Jr., Twisted tensor products. I, Ann. of Math. (2) 69 (1959), 223â246. MR 105687, DOI 10.2307/1970101
- Kuo-Tsai Chen, Extension of $C^{\infty }$ function algebra by integrals and Malcev completion of $\pi _{1}$, Advances in Math. 23 (1977), no. 2, 181â210. MR 458461, DOI 10.1016/0001-8708(77)90120-7
- Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831â879. MR 454968, DOI 10.1090/S0002-9904-1977-14320-6
- Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of KĂ€hler manifolds, Invent. Math. 29 (1975), no. 3, 245â274. MR 382702, DOI 10.1007/BF01389853 V. K. A. M. Gugenheim, On a perturbation theory for the homology of the loop-space, preprint. R. M. Hain, Iterated integrals, minimal models and rational homotopy theory, Ph.D. thesis, Univ. of Illinois, 1980. D. Lehmann, ThĂ©orie homotopique des formes diffĂ©rentielles, AstĂ©risque 45 (1977).
- Jean-Michel Lemaire, AlgĂšbres connexes et homologie des espaces de lacets, Lecture Notes in Mathematics, Vol. 422, Springer-Verlag, Berlin-New York, 1974 (French). MR 0370566
- Jean-Marie Lemaire, ModĂšles minimaux pour les algĂšbres de chaĂźnes, Publ. DĂ©p. Math. (Lyon) 13 (1976), no. 3, 13â26 (French). MR 461500
- John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211â264. MR 174052, DOI 10.2307/1970615
- Joseph Neisendorfer, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. Math. 74 (1978), no. 2, 429â460. MR 494641
- Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205â295. MR 258031, DOI 10.2307/1970725
- D. Sullivan, Differential forms and the topology of manifolds, ManifoldsâTokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 37â49. MR 0370611
- Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Ătudes Sci. Publ. Math. 47 (1977), 269â331 (1978). MR 646078
- Yves FĂ©lix, ModĂšles bifiltrĂ©s: une plaque tournante en homotopie rationnelle, Canadian J. Math. 33 (1981), no. 6, 1448â1458 (French). MR 645238, DOI 10.4153/CJM-1981-111-1
- Richard M. Hain, Iterated integrals and homotopy periods, Mem. Amer. Math. Soc. 47 (1984), no. 291, iv+98. MR 727818, DOI 10.1090/memo/0291
- Daniel TanrĂ©, ModĂšles de Chen, Quillen, Sullivan, Publ. U.E.R. Math. Pures Appl. IRMA 2 (1980), no. 1, exp. no. 2, 87 (French). MR 637192 â, Dualite dâEckmann-Hilton Ă travers les modĂšles de Chen-Quillen-Sullivan, Cahiers Topologie GĂ©om. DiffĂ©rentielle 22 (1981), 53-59.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 277 (1983), 397-411
- MSC: Primary 55P62; Secondary 55U30
- DOI: https://doi.org/10.1090/S0002-9947-1983-0690059-3
- MathSciNet review: 690059