On the action of Steenrod squares on polynomial algebras

Author:
William M. Singer

Journal:
Proc. Amer. Math. Soc. **111** (1991), 577-583

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 well-known 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****[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,*𝐴-generators for certain polynomial algebras*, Math. Proc. Cambridge Philos. Soc.**105**(1989), no. 2, 311–312. MR**974987**, 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**, 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**, 10.1017/S0305004100067797

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1045150-9

Article copyright:
© Copyright 1991
American Mathematical Society