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ArticleOriginal scientific text

Title

The cohomology algebra of certain free loop spaces

Authors 1, 2, 1, 2

Affiliations

  1. Graduate School of Science and Technology, Okayama University, Okayama 700, Japan
  2. Department of Applied Mathematics, Okayama University of Science, 1-1 Ridai-cho, Okayama 700, Japan

Abstract

Let X be a simply connected space and LX the space of free loops on X. We determine the mod p cohomology algebra of LX when the mod p cohomology of X is generated by one element or is an exterior algebra on two generators. We also provide lower bounds on the dimensions of the Hodge decomposition factors of the rational cohomology of LX when the rational cohomology of X is a graded complete intersection algebra. The key to both of these results is the identification of an important subalgebra of the Hochschild homology of a graded complete intersection algebra over a field.

Bibliography

  1. W. Andrzejewski and A. Tralle, Cohomology of some graded differential algebras, Fund. Math. 145 (1994), 181-204.
  2. D. Anick, Connections between Yoneda and Pontrjagin algebras, in: Lecture Notes in Math. 1051, Springer, 1984, 331-350.
  3. D. Burghelea, Z. Fiedorowicz and W. Gajda, Adams operations in Hochschild and cyclic homology of de Rham algebra and free loop spaces, K-Theory 4 (1991), 269-287.
  4. D. Burghelea and M. Vigué-Poirrier, Cyclic homology of commutative algebras I, in: Lecture Notes in Math. 1318, Springer, 1988, 51-72.
  5. S. Eilenberg and J. C. Moore, Homology and fibrations, Comment. Math. Helv. 40 (1966), 199-236.
  6. M. El Haouari, p-Formalité des espaces, J. Pure Appl. Algebra 78 (1992), 27-47.
  7. E. Getzler and J. D. S. Jones, A∞-algebras and the cyclic bar complex, Illinois J. Math. 34 (1990), 256-283.
  8. E. Getzler, J. D. S. Jones and S. Petrack, Differential forms on loop spaces and the cyclic bar complex, Topology 30 (1991), 339-371.
  9. S. Halperin and J. Stasheff, Obstructions to homotopy equivalences, Adv. Math. 32 (1979), 233-279.
  10. S. Halperin and M. Vigué-Poirrier, The homology of a free loop space, Pacific J. Math. 147 (1991), 311-324.
  11. K. Kuribayashi, On the mod p cohomology of the spaces of free loops on the Grassmann and Stiefel manifolds, J. Math. Soc. Japan 43 (1991), 331-346.
  12. RD. L. Rector, Steenrod operations in the Eilenberg-Moore spectral sequence, Comment. Math. Helv. 45 (1970), 540-552.
  13. L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), 58-93.
  14. L. Smith, On the Künneth theorem I, Math. Z. 166 (1970), 94-140.
  15. L. Smith, On the characteristic zero cohomology of the free loop space, Amer. J. Math. 103 (1981), 887-910.
  16. L. Smith, A note on the realization of graded complete intersection algebras by the cohomology of a space, Quart. J. Math. Oxford Ser. (2) 33 (1982), 379-384.
  17. L. Smith, The Eilenberg-Moore spectral sequence and the mod 2 cohomology of certain free loop spaces, Illinois J. Math. 28 (1984), 516-522.
  18. M. Vigué-Poirrier and D. Burghelea, A model for cyclic homology and algebraic K-theory of 1-connected topological spaces, J. Differential Geom. 22 (1985), 243-253.
  19. M. Vigué-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J. Differential Geom. 11 (1976), 633-644.
Fundamenta Mathematicae
1997/154/1
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Pages:
57-73
Main language of publication
English
Received
1996-05-23
Accepted
1997-01-21
Published
1997
Exact and natural sciences