On the realization and classification of symmetric algebras as cohomology rings
Author:
Larry Smith
Journal:
Proc. Amer. Math. Soc. 87 (1983), 144148
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 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.
 [1]
J.
F. Adams and C.
W. Wilkerson, Finite 𝐻spaces and algebras over the
Steenrod algebra, Ann. of Math. (2) 111 (1980),
no. 1, 95–143. MR 558398
(81h:55006), http://dx.doi.org/10.2307/1971218
 [2]
Jaume
Aguadé, Cohomology algebras with two generators, Math.
Z. 177 (1981), no. 2, 289–296. MR 612881
(82d:55003), http://dx.doi.org/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
(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 Mathematics, Vol. 304, SpringerVerlag, Berlin, 1972. MR 0365573
(51 #1825)
 [6]
Allan
Clark and John
Ewing, The realization of polynomial algebras as cohomology
rings, Pacific J. Math. 50 (1974), 425–434. MR 0367979
(51 #4221)
 [7]
Daniel
Quillen, On the cohomology and 𝐾theory of the general
linear groups over a finite field, Ann. of Math. (2)
96 (1972), 552–586. MR 0315016
(47 #3565)
 [8]
Louis
Solomon, Invariants of finite reflection groups, Nagoya Math.
J. 22 (1963), 57–64. MR 0154929
(27 #4872)
 [9]
Larry
Smith, Homological algebra and the
EilenbergMoore spectral sequence, Trans. Amer.
Math. Soc. 129
(1967), 58–93. MR 0216504
(35 #7337), http://dx.doi.org/10.1090/S00029947196702165046
 [10]
Larry
Smith, The cohomology of stable two stage Postnikov systems,
Illinois J. Math. 11 (1967), 310–329. MR 0208597
(34 #8406)
 [11]
Larry
Smith, On the characteristic zero cohomology of the free loop
space, Amer. J. Math. 103 (1981), no. 5,
887–910. MR
630771 (83k:57035), http://dx.doi.org/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), 379384.
 [13]
, The homotopy classification of maps between certain spaces with polynomial cohomology, Göttingen Univ. Preprint SS1981.
 [14]
L. Smith and R. M. Switzer, Polynomial algebras over the Steenrod algebra, Variations on a theme of Adams and Wilkerson (to appear).
 [1]
 J. F. Adams and C. W. Wilkerson, Finite spaces and algebras over the Steenrod algebra, Ann. of Math. (2) 111 (1980), 95143. MR 558398 (81h:55006)
 [2]
 J. Aguadé, Cohomology algebras with two generation, Math. Z. 177 (1981), 289294. MR 612881 (82d:55003)
 [3]
 P. F. Baum and L. Smith, The real cohomology of differential fiber bundles, Comment. Math. Helv. 42 (1967), 171179. 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, SpringerVerlag, 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), 425434. 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), 552586. MR 0315016 (47:3565)
 [8]
 L. Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 5764. MR 0154929 (27:4872)
 [9]
 L. Smith, Homological algebra and the EilenbergMoore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), 5893. MR 0216504 (35:7337)
 [10]
 , Cohomology of stable two stage Postnikov systems. Illinois J. Math. 11 (1967), 310329. MR 0208597 (34:8406)
 [11]
 , On the rational cohomology of the free loop space, Amer. J. Math. 103 (1981), 887910. 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), 379384.
 [13]
 , The homotopy classification of maps between certain spaces with polynomial cohomology, Göttingen Univ. Preprint SS1981.
 [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:
http://dx.doi.org/10.1090/S00029939198306772502
PII:
S 00029939(1983)06772502
Article copyright:
© Copyright 1983 American Mathematical Society
