manuscripta mathematica

, Volume 58, Issue 3, pp 363–376 | Cite as

Applications of perturbation theory to iterated fibrations

  • Larry Lambe
  • Jim Stasheff


For a class of spaces including simply connected spaces and classifying spaces of nilpotent groups, relatively small differential graded algebras are constructed over commutative rings with 1 which are chain homotopy equivalent to the singular cochain algebra. An application to finitely generated torsion-free nilpotent groups over the integers is given.


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  1. [1]
    BARCUS, W.D.: On the complexes of Hirsch and Eilenberg-Moore, Proc. Camb. Phil. Soc.64, 953–960(1968)Google Scholar
  2. [2]
    BARCUS, W.D.: Principal bundles and Hirsch's spectral sequence, Quart. J. Math., Oxford17, 321–334(1966)Google Scholar
  3. [3]
    BARRETT, M.G.: V.K.A.M. Gugenheim and J.C. Moore: On simisimplicial fibre-bundles, Amer. J. Math.81, 639–657(1959)Google Scholar
  4. [4]
    BROWN, E.H. Jr.: Twisted tensor products, Ann. Math.1, 223–246(1959)Google Scholar
  5. [5]
    BROWN, R.: The twisted Eilenberg-Zilber theorem, Celebrazioni Archimedee del secolo XX, Simposio di topologia 34–37(1967)Google Scholar
  6. [6]
    CARTAN, H.: Séminaire Cartan (École Norm. Sup., Paris) 1954–5, Algebrès d'Eilenberg-MacLane et Homotopie, exposés 2–11Google Scholar
  7. [7]
    CHEN, K-T.: On the composition functions of nilpotent Lie groups, Proc. Amer. Math. Soc.8, 1158–1159(1957)Google Scholar
  8. [8]
    COCKCROFT, W.H.: The cohomology groups of a fibre space with fibre a space of type K(π,n), Proc. Amer. Math. Soc.7, 1120–1126(1956)Google Scholar
  9. [9]
    COCKCROFT, W.H.: The cohomology groups of a fibre space with fibre a space of type, K(π,n), II, Trans. Amer. Math. Soc.91, 505–524(1959)Google Scholar
  10. [10]
    COCKCROFT, W.H.: On a theorem of Borel, Trans. Amer. Math. Soc.98, 255–262(1961)Google Scholar
  11. [11]
    EILENBERG, S. and MACLANE, S.: On the groups H(π,n) I, Ann. Math.58, 55–106(1953)Google Scholar
  12. [12]
    EILENBERG, S. and MACLANE, S.: On the groups H(π,n) II, Ann. Math.60, 49–139(1954)Google Scholar
  13. [13]
    GUGENHEIM, V.K.A.M. and STASHEFF, J.: On perturbations andA structures, Bull. Soc. Math. de Belg. (to appear)Google Scholar
  14. [14]
    GUGENHEIM, V.K.A.M.: On a theorem of E.H. Brown, IL. J. Math.4, 292–311(1960)Google Scholar
  15. [15]
    GUGENHEIM, V.K.A.M.: On a the chain complex of a fibration, IL. J. Math.3, 398–414(1972)Google Scholar
  16. [16]
    GUGENHEIM, V.K.A.M.: On a perturbation theory for the homology of the loop-space, J. Pure & Appl. Alg.25, 197–205(1982)Google Scholar
  17. [17]
    GUGENHEIM, V.K.A.M. and MAY, J.P.: On the theory and application of differential torsion products, Mems. Am. Math. Soc. 142(1974)Google Scholar
  18. [18]
    GUGENHEIM, V.K.A.M. and MUNKHOLM, H.J.: On the extended functoriality of Tor and Cotor, J. Pure & Appl. Alg.4, 9–29(1974)Google Scholar
  19. [19]
    HILTON, P.: Homotopy theory and duality, Notes on Mathematics and its applications, Gordon and Breach, N.Y.Google Scholar
  20. [20]
    HALL, P.: Nilpotent groups, Canad. Math. Cong., Edmonton, 1957Google Scholar
  21. [21]
    HALPERIN, S. and STASHEFF, J.D.: Differential algebra in its own rite, Proc. Adv. Study Inst. Top., Aarhus, 1970Google Scholar
  22. [22]
    HIRSCH, GUY: Sur les groups d'homologies des espaces fibres, Bull. Soc. Math. de Belg.6, (1953)79–96Google Scholar
  23. [23]
    HUSEMOLLER, D., MOORE, J. and Stasheff, J.D.: Differential homological algebra and homogeneous spaces, J. Pure & Appl. Alg.5, 113–185 (1974)Google Scholar
  24. [24]
    HÜBSCHMANN, J.: The homotopy type ofFψq, the complex and symplectic cases, Cont. Math.55, 487–518(1986)Google Scholar
  25. [25]
    LAMBE, L.: Cohomology of principal G-bundles over a torus whenH *(BG;R) is polynomial, Bull. Soc. Math. de Belg. (to appear)Google Scholar
  26. [26]
    LAMBE, L. and PRIDDY, S.: Cohomology of nilmanifolds and torsionfree nilpotent groups, Trans. Amer. Math. Soc.273, 39–55(1982)Google Scholar
  27. [27]
    MACLANE, S.: Homology, Die Grundlehren der Math. Wissenschaften, Band 114, Springer-Verlag, N.Y.Google Scholar
  28. [28]
    MAY, PETER J.: Simplicial objects in Algebraic topology, Mathematical studies #11, Van Nostrand, 1967Google Scholar
  29. [29]
    MOORE, J.C. Séminaire Cartan (École Norm. Sup., Paris) 1954–5, Comparaison de la Bar-construction à la construction W et aux complexesK(π,n), exposés 13Google Scholar
  30. [30]
    QUILLEN, D.: Rational homotopy theory, Ann. Math.90, 205–195(1969).Google Scholar
  31. [31]
    SZCZARBA, R.H.: The homology of twisted cartesian products, Trans. Am. Math. Soc.100, 197–216(1961)Google Scholar
  32. [32]
    SHIH, W.: Homology des espaces fibre, Inst. des Hautes Études Sci.13, 93–176(1962)Google Scholar
  33. [33]
    STASHEFF, J.: Homotopy associativity of H-spaces II, Trans. Amer. Math. Soc.108, 293–312(1963)Google Scholar
  34. [34]
    SULLIVAN, D.: Infinitesimal computations in topology, Publ. Inst. Hautes Études Sci.47, 269–331(1977)Google Scholar
  35. [35]
    WALL, C.T.C.: Resolutions for extensions of groups, Proc. Camb. Phil. Soc.57, 251–255(1961)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Larry Lambe
    • 1
    • 2
  • Jim Stasheff
    • 1
    • 2
  1. 1.Department of MathematicsNotre Dame UniversityNotre Dame
  2. 2.Department of MathematicsUniversity of North CarolinaChapel Hill

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