Free loop spaces of finite complexes have infinite category
Authors:
Y. Félix, J. C. Thomas and M. Vigué-Poirrier
Journal:
Proc. Amer. Math. Soc. 111 (1991), 869-875
MSC:
Primary 55P50; Secondary 55P62
DOI:
https://doi.org/10.1090/S0002-9939-1991-1025277-8
MathSciNet review:
1025277
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be a
-connected space such that each
is finitely generated. In this paper we prove that if the reduced homology of
with coefficients in a field is nonzero, then the Lusternik-Schnirelmann category of the free loop space is infinite.
- [1] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. Math. 57 (1953), 115-207. MR 0051508 (14:490e)
- [2] E. Fadell and S. Husseini, A note on the category of the free loop space, Proc. Amer. Math. Soc. 107 (1989), 527-536. MR 984789 (90a:55008)
- [3] Y. Félix and S. Halperin, Rational L. S. category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1-37. MR 664027 (84h:55011)
- [4]
Y. Félix, S. Halperin, J. M. Lemaire, and J. C. Thomas,
loop space homology, Invent. Math. 95 (1989), 247-262. MR 974903 (89k:55010)
- [5] Y. Félix, S. Halperin, and J. C. Thomas, Gorenstein spaces, Adv. in Math. 71 (1988), 92-112. MR 960364 (89k:55019)
- [6] -, Hopf algebras of polynomial growth, J. Algebra 125 (1989), 408-417. MR 1018954 (90j:16021)
- [7] Y. Félix, Engel elements in the homotopy Lie algebra, preprint 1989.
- [8] -, Elliptic Hopf algebras, preprint 1989.
- [9] J. McCleary, Homotopy theory and closed geodesic, preprint 1989. MR 1048178 (91e:57060)
- [10] J. C. Moore and L. Smith, Hopf algebras and multiplicative fibrations I, Amer. J. Math. 90 (1968), 752-780. MR 0234455 (38:2772)
- [11] -, Hopf algebras and multiplicative fibrations II, Amer. J. Math. 90 (1968), 1113-1150. MR 0238323 (38:6599)
- [12] J.-E. Roos, Homology of free loop spaces, cyclic homology, and nonrationnal Poincaré-Betti series in commutative algebra, Reports, Univ. of Stockholm, 1988; Lecture Notes in Math. (to appear). MR 981826 (90f:55020)
- [13] D. Sullivan, Differential forms and the topology of manifolds, Proc. of the Intern. Conf. on Manifolds (Tokyo, 1973) (Akio Hattori, ed.), Tokyo Univ. Press, Tokyo, 1975, pp. 37-49. MR 0370611 (51:6838)
- [14] D. Sullivan and M. Vigué-Poirrier, The homology theory of the closed geodesic problem, J. Differential Geom. 11 (1976), 633-644. MR 0455028 (56:13269)
- [15] M. Vigué-Poirrier, Dans le fibré de l'espace des lacets libres, la fibre n'est pas, en général, totalement non cohomologue à zéro, Math. Z. 181 (1982), 537-542. MR 682673 (84e:55007)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1991-1025277-8
Article copyright:
© Copyright 1991
American Mathematical Society