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 that is an unstable algebra over the Steenrod algebra, an odd prime, that satisfies

**[1]**J. F. Adams and C. W. Wilkerson,*Finite**-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**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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0677250-2

Article copyright:
© Copyright 1983
American Mathematical Society