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Transactions of the American Mathematical Society

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Hochschild homology criteria for trivial algebra structures

Author: Micheline Vigué-Poirrier
Journal: Trans. Amer. Math. Soc. 354 (2002), 3869-3882
MSC (2000): Primary 13N05, 18F25, 55P62
Published electronically: June 4, 2002
MathSciNet review: 1926856
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Abstract: We prove two similar results by quite different methods. The first one deals with augmented artinian algebras over a field: we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the Hochschild homology (or cyclic homology) groups. For the second result, let $X$ be a 1-connected finite CW-complex. We characterize the trivial algebra structure on the cohomology algebra of $X$ with coefficients in a fixed field in terms of the maximality of the Betti numbers of the free loop space.

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  • [An] D. Anick, A model of Adams Hilton type for fibre squares, Illinois J. Math., 29, (1985), 463-502. MR 86h:55009
  • [A-H] J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv., 30, (1956), 305-330. MR 17:1119b
  • [B-L] H. Baues and J. M. Lemaire, Minimal models in homotopy theory, Math. Ann., 225, (1977), 219-242. MR 55:4174
  • [B-F] D. Burghelea and Z. Fiedorowicz, Cyclic homology and algebraic $K$-theory of spaces, Topology, 25, (1986), 303-317. MR 88i:18009b
  • [B-V] D. Burghelea and M. Vigué-Poirrier, Cyclic homology of commutative algebras, Lecture Notes in Mathematics, 1318, (1988), 51-72. MR 89k:18027
  • [E-H] El Haouari, $P$-Formalité des espaces, J. Pure Appl. Algebra, 78, (1992), 27-47. MR 93d:55016
  • [Go] T. Goodwillie, Cyclic homology, derivations and the free loop space, Topology, 24, (1985), 187-215. MR 87c:18009
  • [G-M] D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Differential Geom., 3, (1969), 493-510. MR 41:9143
  • [Jo] J. Jones, Cyclic homology and equivariant homology, Invent. Math., 87, (1987), 403-423. MR 88f:18016
  • [H-S] S. Halperin and J. Stasheff, Obstructions to homotopy equivalences, Advances in Math., 32, (1979), 233-279. MR 80j:55016
  • [H-V] S. Halperin and M. Vigué-Poirrier, The homology of a free loop space, Pacific J. Math., 147, (1991), 311-324. MR 92e:55012
  • [H-K-R] G. Hochschild, B. Kostant, and A. Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc., 102, (1962), 383-408. MR 26:167
  • [La1] P. Lambrechts, Analytic properties of Poincaré series of spaces, Topology, 37, (1998), 1363-1370. MR 99e:55011
  • [La2] P. Lambrechts, On the Betti numbers of the free loop space of a coformal space, J. Pure Appl. Algebra, 161, (2001), 177-192. MR 2002d:55015
  • [Lo] J. L. Loday, Cyclic homology, Springer-Verlag, Berlin, (1992). MR 94a:19004
  • [L-Q] J. L. Loday and D. Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv., 59, (1984), 565-591. MR 86i:17003
  • [Pa] M. Parhizgar, On the cohomology ring of the free loop space of a wedge of spheres, Math. Scand, 80, (1997), 195-248. MR 98i:55014
  • [Ro] J. E. Roos, Homology of free loop spaces, cyclic homology, and nonrational Poincaré-Betti series in commutative algebra, Lecture Note in Math., 1352, (1988), 173-189. MR 90f:55020
  • [Vi1] M. Vigué-Poirrier, Homotopie rationnelle et croissance du nombre de géodésiques fermées, Ann. Sci. Ecole Norm. Sup., 17, (1984), 413-431. MR 86h:58027
  • [Vi2] M. Vigué-Poirrier, Homologie de Hochschild et homologie cyclique des algèbres différentielles graduées, Astérisque 191, (1990), 255-267. MR 92e:19003

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Additional Information

Micheline Vigué-Poirrier
Affiliation: Université Paris-Nord, Institut Galilée, Département de Mathématiques, F-93430 Villetaneuse, France

Keywords: Augmented algebra, Hochschild homology, cyclic homology, free loop space, minimal model of a differential graded algebra
Received by editor(s): March 23, 2001
Received by editor(s) in revised form: March 15, 2002
Published electronically: June 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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