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On the realization and classification of symmetric algebras as cohomology rings


Author: Larry Smith
Journal: Proc. Amer. Math. Soc. 87 (1983), 144-148
MSC: Primary 57T15; Secondary 55N99
DOI: https://doi.org/10.1090/S0002-9939-1983-0677250-2
MathSciNet review: 677250
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Abstract: In this note, we prove that every algebra of the form $ E[\sigma , \ldots ,{\sigma _n}] \otimes P[{\rho _1}, \ldots ,{\rho _n}]$ that is an unstable algebra over the $ \mod p$ Steenrod algebra, $ p$ an odd prime, that satisfies

$\displaystyle \beta {\sigma _i} = {\rho _i},\quad \deg \rho_i \not\equiv 0(p),\quad i = 1, \ldots ,n,$

arises as the cohomology of at least one topological space. Moreover, we show that the classification of such algebras is implicit in the work of Adams and Wilkerson and Clark and Ewing.

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

  • [1] J. F. Adams and C. W. Wilkerson, Finite $ H$-spaces and algebras over the Steenrod algebra, Ann. of Math. (2) 111 (1980), 95-143. MR 558398 (81h:55006)
  • [2] J. Aguadé, Cohomology algebras with two generation, Math. Z. 177 (1981), 289-294. MR 612881 (82d:55003)
  • [3] P. F. Baum and L. Smith, The real cohomology of differential fiber bundles, Comment. Math. Helv. 42 (1967), 171-179. MR 0221522 (36:4574)
  • [4] N. Bourbaki, Groupes et algèbres de Lie, Chapter V, Hermann, Paris, 1962.
  • [5] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, Berlin and New York, 1972. MR 0365573 (51:1825)
  • [6] A. Clark and J. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425-434. MR 0367979 (51:4221)
  • [7] D. Quillen, On the cohomology and $ K$ theory of general linear groups over finite fields, Ann. of Math. (2) 96 (1972), 552-586. MR 0315016 (47:3565)
  • [8] L. Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57-64. MR 0154929 (27:4872)
  • [9] L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), 58-93. MR 0216504 (35:7337)
  • [10] -, Cohomology of stable two stage Postnikov systems. Illinois J. Math. 11 (1967), 310-329. MR 0208597 (34:8406)
  • [11] -, On the rational cohomology of the free loop space, Amer. J. Math. 103 (1981), 887-910. MR 630771 (83k:57035)
  • [12] -, A note on the realization of graded intersection algebras as the cohomology of a space, Quart. J. Math. Oxford Ser. (2) 83 (1982), 379-384.
  • [13] -, The homotopy classification of maps between certain spaces with polynomial cohomology, Göttingen Univ. Preprint SS-1981.
  • [14] L. Smith and R. M. Switzer, Polynomial algebras over the Steenrod algebra, Variations on a theme of Adams and Wilkerson (to appear).

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DOI: https://doi.org/10.1090/S0002-9939-1983-0677250-2
Article copyright: © Copyright 1983 American Mathematical Society

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