Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the evaluation map

Author: Aniceto Murillo
Journal: Trans. Amer. Math. Soc. 339 (1993), 611-622
MSC: Primary 55P62; Secondary 18G15
MathSciNet review: 1112376
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The evaluation map of a differential graded algebra or of a space is described under two different approaches. This concept turns out to have geometric implications: (i) A $ 1$-connected topological space, with finite-dimensional rational homotopy, has finite-dimensional rational cohomology if and only if it has nontrivial evaluation map. (ii) Let $ E\xrightarrow{\rho }B$ be a fibration of simplyconnected spaces. If the rational cohomology of the fibre is finite dimensional and the evaluation map of the base is different from zero, then the evaluation map of the total space is nonzero. Also, if $ \rho $ is surjective in rational homotopy and the evaluation map of $ E$ is nontrivial, then the evaluation map of the fibre is different from zero.

References [Enhancements On Off] (What's this?)

  • [1] Y. Félix and S. Halperin, Rational $ L{\text{-}}S$ category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1-37. MR 664027 (84h:55011)
  • [2] Y. Félix, S. Halperin, J. M. Lemairè, and J. C. Thomas, $ \operatorname{Mod} p$ loop space homology, Invent. Math. 92 (1989), 247-262. MR 974903 (89k:55010)
  • [3] Y. Félix, S. Halperin, and J. C. Thomas, Gorenstein spaces, Adv. in Math. 71 (1988), 92-112. MR 960364 (89k:55019)
  • [4] W. Greub, S. Halperin, and R. Vanstone, Connections, curvature, and cohomology, vol. III, Academic Press, New York, 1972. MR 0400275 (53:4110)
  • [5] S. Halperin, Lectures on minimal models, Mém. Soc. Math. France 9/10 (1983). MR 736299 (85i:55009)
  • [6] -, Torsion gaps for finite complexes. II, Topology 30 (1991), 471-478. MR 1113690 (92f:55014)
  • [7] A. Murillo, Rational fibrations in differential homological algebra, Trans. Amer. Math. Soc. 332 (1992), 79-91. MR 1079055 (92j:55019)
  • [8] L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1970), 58-93. MR 0216504 (35:7337)
  • [9] E. H. Spanier, Homology theory of fiber bundles, Proc. Internat. Congr. Math., Cambridge, Mass., 1950, pp. 390-396. MR 0044838 (13:486b)
  • [10] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1978), 269-331. MR 0646078 (58:31119)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55P62, 18G15

Retrieve articles in all journals with MSC: 55P62, 18G15

Additional Information

Keywords: Evaluation map, rational homotopy, Sullivan models, elliptic spaces, Partially supported by a DGICYT grant (PB88-0329)
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society