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.
@article{bwmeta1.element.bwnjournal-article-fmv154i1p57bwm,
author = {Toshihiro Yamaguchi and Katsuhiko Kuribayashi},
title = {The cohomology algebra of certain free loop spaces},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {57-73},
zbl = {0883.55006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv154i1p57bwm}
}
Yamaguchi, Toshihiro; Kuribayashi, Katsuhiko. The cohomology algebra of certain free loop spaces. Fundamenta Mathematicae, Tome 154 (1997) pp. 57-73. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv154i1p57bwm/
[00000] [1] W. Andrzejewski and A. Tralle, Cohomology of some graded differential algebras, Fund. Math. 145 (1994), 181-204. | Zbl 0847.55007
[00001] [2] D. Anick, Connections between Yoneda and Pontrjagin algebras, in: Lecture Notes in Math. 1051, Springer, 1984, 331-350.
[00002] [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. | Zbl 0726.55007
[00003] [4] D. Burghelea and M. Vigué-Poirrier, Cyclic homology of commutative algebras I, in: Lecture Notes in Math. 1318, Springer, 1988, 51-72.
[00004] [5] S. Eilenberg and J. C. Moore, Homology and fibrations, Comment. Math. Helv. 40 (1966), 199-236. | Zbl 0148.43203
[00005] [6] M. El Haouari, p-Formalité des espaces, J. Pure Appl. Algebra 78 (1992), 27-47.
[00006] [7] E. Getzler and J. D. S. Jones, -algebras and the cyclic bar complex, Illinois J. Math. 34 (1990), 256-283. | Zbl 0701.55009
[00007] [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. | Zbl 0729.58004
[00008] [9] S. Halperin and J. Stasheff, Obstructions to homotopy equivalences, Adv. Math. 32 (1979), 233-279. | Zbl 0408.55009
[00009] [10] S. Halperin and M. Vigué-Poirrier, The homology of a free loop space, Pacific J. Math. 147 (1991), 311-324. | Zbl 0666.55011
[00010] [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. | Zbl 0729.55010
[00011] [12] RD. L. Rector, Steenrod operations in the Eilenberg-Moore spectral sequence, Comment. Math. Helv. 45 (1970), 540-552. | Zbl 0209.27501
[00012] [13] L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), 58-93. | Zbl 0177.51402
[00013] [14] L. Smith, On the Künneth theorem I, Math. Z. 166 (1970), 94-140. | Zbl 0189.54401
[00014] [15] L. Smith, On the characteristic zero cohomology of the free loop space, Amer. J. Math. 103 (1981), 887-910. | Zbl 0475.55004
[00015] [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. | Zbl 0546.55009
[00016] [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. | Zbl 0538.57025
[00017] [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. | Zbl 0595.55009
[00018] [19] M. Vigué-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J. Differential Geom. 11 (1976), 633-644. | Zbl 0361.53058