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

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 𝐻-spaces and algebras over the Steenrod algebra*, Ann. of Math. (2)**111**(1980), no. 1, 95–143. MR**558398**, 10.2307/1971218**[2]**Jaume Aguadé,*Cohomology algebras with two generators*, Math. Z.**177**(1981), no. 2, 289–296. MR**612881**, 10.1007/BF01214207**[3]**Paul Baum and Larry Smith,*The real cohomology of differentiable fibre bundles*, Comment. Math. Helv.**42**(1967), 171–179. MR**0221522****[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 Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR**0365573****[6]**Allan Clark and John Ewing,*The realization of polynomial algebras as cohomology rings*, Pacific J. Math.**50**(1974), 425–434. MR**0367979****[7]**Daniel Quillen,*On the cohomology and 𝐾-theory of the general linear groups over a finite field*, Ann. of Math. (2)**96**(1972), 552–586. MR**0315016****[8]**Louis Solomon,*Invariants of finite reflection groups*, Nagoya Math. J.**22**(1963), 57–64. MR**0154929****[9]**Larry Smith,*Homological algebra and the Eilenberg-Moore spectral sequence*, Trans. Amer. Math. Soc.**129**(1967), 58–93. MR**0216504**, 10.1090/S0002-9947-1967-0216504-6**[10]**Larry Smith,*The cohomology of stable two stage Postnikov systems*, Illinois J. Math.**11**(1967), 310–329. MR**0208597****[11]**Larry Smith,*On the characteristic zero cohomology of the free loop space*, Amer. J. Math.**103**(1981), no. 5, 887–910. MR**630771**, 10.2307/2374251**[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:
http://dx.doi.org/10.1090/S0002-9939-1983-0677250-2

Article copyright:
© Copyright 1983
American Mathematical Society