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Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 30 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
Differential Operators and the Steenrod Algebra
, 1995
"... This article presents an elementary treatment of the Steenrod algebra from an algebraic point of view in terms of differential operators acting on polynomials. The exposition concentrates on the Steenrod algebra A over the field F 2 of two elements although the approach works for the odd prime field ..."
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Cited by 9 (3 self)
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This article presents an elementary treatment of the Steenrod algebra from an algebraic point of view in terms of differential operators acting on polynomials. The exposition concentrates on the Steenrod algebra A over the field F 2 of two elements although the approach works for the odd prime fields. From a
New relationships among loopspaces, symmetric products, and Eilenberg MacLane spaces
 in Cohomological Methods in Homotopy Theory (Barcelona, 1998), Birkhauser Verlag Progress in Math
"... Abstract. Let T (j) be the dual of the jth stable summand of Ω2S3 (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T (j) ! T (2j) ! : : : is an innite wedge of stable summands of K(V ..."
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Cited by 4 (1 self)
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Abstract. Let T (j) be the dual of the jth stable summand of Ω2S3 (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T (j) ! T (2j) ! : : : is an innite wedge of stable summands of K(V; 1)’s, where V denotes an elementary abelian 2 group. In particular, when one starts with T (1), one gets K(Z=2; 1) = RP1 as one of the summands. I discuss a generalization of this picture using higher iterated loopspaces and Eilenberg MacLane spaces. I consider certain nite spectra T (n; j) for n; j 0 (with T (1; j) = T (j)), dual to summands of Ωn+1SN, conjecture generalizations of the above, and prove that these conjectures are correct in cohomology. So, for example, T (n; j) has unstable cohomology, and the cohomology of the hocolimit of a certain sequence T (n; j) ! T (n; 2j) ! : :: agrees with the cohomology of the wedge of stable summands of K(V; n)’s corresponding to the wedge occurring in the n = 1 case above. One can also map the T (n; j) to each other as n varies, and here the coho
New cohomological relationships among loopspaces, symmetric products, and Eilenberg MacLane spaces
, 1996
"... Abstract. Let T (j) be the dual of the jth BrownGitler spectrum (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, is dual to a stable summand of Ω2S3, and that the homotopy colimit of a certain sequence T (j) ! T (2j) ! : : : is a wed ..."
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Cited by 3 (3 self)
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Abstract. Let T (j) be the dual of the jth BrownGitler spectrum (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, is dual to a stable summand of Ω2S3, and that the homotopy colimit of a certain sequence T (j) ! T (2j) ! : : : is a wedge of stable summands of K(V; 1)’s, where V denotes an elementary abelian 2 group. In particular, when one starts with T (1), one gets K(Z=2; 1) = RP1 as one of the summands. Re ning a question posed by Doug Ravenel, I discuss a generalization of this picture. I consider certain nite spectra T (n; j) for n; j 0 (with T (1; j) = T (j)), dual to summands of Ωn+1SN, conjecture generalizations of all of the above, and prove that all these conjectures are correct in cohomology. So, for example, T (n; j) has unstable cohomology, and the cohomology of the colimit of a certain sequence T (n; j) ! T (n; 2j) ! : : : agrees with the cohomology of the wedge of stable summands of K(V; n)’s corresponding to the wedge occurring in the n = 1 case above.
Contents
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may nd of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of ..."
Abstract
 Add to MetaCart
This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may nd of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below.
COHOMOLOGY OF STEENROD, NEUTRALITY OF REAL PROJECTIVE SPACE
"... algebra on the Steenrod squares Abstract. Let A χ− → A be the canonical antiautomorphism of the mod 2 Steenrod algebra A. For s ≥ 1 let Ad(s) be the set of all the admissible monomials Sqi1 · · ·Sqis that appear in the admissible expansion of χ(Sqi) for some i. In this paper we show Ad(s) can be ..."
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algebra on the Steenrod squares Abstract. Let A χ− → A be the canonical antiautomorphism of the mod 2 Steenrod algebra A. For s ≥ 1 let Ad(s) be the set of all the admissible monomials Sqi1 · · ·Sqis that appear in the admissible expansion of χ(Sqi) for some i. In this paper we show Ad(s) can be determined from the admissible expansion of