On the action of Steenrod squares on polynomial algebras
Author:
William M. Singer
Journal:
Proc. Amer. Math. Soc. 111 (1991), 577583
MSC:
Primary 55S10; Secondary 55Q10, 55Q40, 55S05, 55T15
MathSciNet review:
1045150
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Abstract: Let be the cohomology of the elementary abelian group ( factors). The Steenrod algebra acts on according to wellknown rules. If denotes the augmentation ideal, then we are interested in determining the image of the action : the space of elements in that are hit by positive dimensional Steenrod squares. The problem is motivated by applications to cobordism theory [P1] and the homology of the Steenrod algebra [S]. Our main result, which generalizes work of Wood [W], identifies a new class of hit monomials.
 [A]
J.
F. Adams, On formulae of Thom and Wu, Proc. London Math. Soc.
(3) 11 (1961), 741–752. MR 0139177
(25 #2613)
 [BP]
E.
H. Brown Jr. and F.
P. Peterson, 𝐻*(𝑀𝑂) as an algebra over the
Steenrod algebra, Conference on homotopy theory (Evanston, Ill., 1974)
Notas Mat. Simpos., vol. 1, Soc. Mat. Mexicana, México, 1975,
pp. 11–19. MR
761717
 [P1]
Franklin
P. Peterson, 𝐴generators for certain polynomial
algebras, Math. Proc. Cambridge Philos. Soc. 105
(1989), no. 2, 311–312. MR 974987
(90a:55031), http://dx.doi.org/10.1017/S0305004100067803
 [P2]
, Generators of as a module over the Steenrod algebra, Abstracts Amer. Math. Soc., no. 833, April 1987.
 [S]
William
M. Singer, The transfer in homological algebra, Math. Z.
202 (1989), no. 4, 493–523. MR 1022818
(90i:55035), http://dx.doi.org/10.1007/BF01221587
 [W]
R.
M. W. Wood, Steenrod squares of polynomials and the Peterson
conjecture, Math. Proc. Cambridge Philos. Soc. 105
(1989), no. 2, 307–309. MR 974986
(90a:55030), http://dx.doi.org/10.1017/S0305004100067797
 [A]
 J. F. Adams, On formulae of Thom and Wu, Proc. London Math. Soc. 11 (1961). MR 0139177 (25:2613)
 [BP]
 E. Brown and F. P. Peterson, as an algebra over the Steenrod algebra, Notas Mat. Simpos. 1 (1975), 1121. MR 761717
 [P1]
 F. P. Peterson, generators for certain polynomial algebras, Math. Proc. Cambridge Philos. Soc. 105 (1989), 311312. MR 974987 (90a:55031)
 [P2]
 , Generators of as a module over the Steenrod algebra, Abstracts Amer. Math. Soc., no. 833, April 1987.
 [S]
 W. M. Singer, The transfer in homological algebra, Math. Z. 202 (1989), 493523. MR 1022818 (90i:55035)
 [W]
 R. M. W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Philos. Soc. 105 (1989), 307309. MR 974986 (90a:55030)
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DOI:
http://dx.doi.org/10.1090/S00029939199110451509
PII:
S 00029939(1991)10451509
Article copyright:
© Copyright 1991
American Mathematical Society
