Cohomology operations and the Steenrod algebra

2008 ◽  
pp. 161-180
Keyword(s):  
Steenrod Algebra ◽  
Cohomology Operations

On the differentials in the Adams spectral sequence

1964 ◽  
Vol 60 (3) ◽  
pp. 409-420 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Spectral Sequence ◽  
Higher Order ◽  
Adams Spectral Sequence ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Cohomology Operations ◽  
Image Position

In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements ofcorresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.

Hit polynomials and conjugation in the dual Steenrod algebra

1998 ◽  
Vol 123 (3) ◽  
pp. 531-547 ◽  
Author(s):  
JUDITH H. SILVERMAN
Keyword(s):  
Steenrod Algebra ◽  
Augmentation Ideal ◽  
Cohomology Operations ◽  
Positive Integers

Let [Ascr ]* be the mod-2 Steenrod algebra of cohomology operations and χ its canonical antiautomorphism. For all positive integers f and k, we show that the excess of the element χ[Sq (2k−1f)· Sq (2k−2f)… Sq (2f)·Sq (f)] is (2k−1)μ(f), where μ(f) denotes the minimal number of summands in any representation of f as a sum of numbers of the form 2i−1. We also interpret this result in purely combinatorial terms. In so doing, we express the Milnor basis representation of the products Sq (a1)…Sq (an) and χ[Sq (a1)…Sq (an)] in terms of the cardinalities of certain sets of matrices.For s[ges ]1, let ℙs=[ ]2= [x1, …, xs] be the mod-2 cohomology of the s-fold product of ℝP∞ with itself, with its usual structure as an [Ascr ]*-module. A polynomial P∈ℙs is hit if it is in the image of the action [Ascr ]*×ℙs→ℙs, where [Ascr ]* is the augmentation ideal of [Ascr ]*. We prove that if the integers e, f, and k satisfy e<(2k−1)μ(f), then for any polynomials E and F of degrees e and f respectively, the product E·F2k is hit. This generalizes a result of Wood conjectured by Peterson, and proves a conjecture of Singer and Silverman.

Frame Fields on Manifolds

1986 ◽  
Vol 38 (1) ◽  
pp. 232-256 ◽  
Author(s):  
Tze Beng Ng
Keyword(s):  
Steenrod Algebra ◽  
Cohomology Operation ◽  
Cohomology Operations ◽  
Image Position

Consider the following stable secondary cohomology operations associated with the relations in the mod 2 Steenrod algebra: such thatLet ψ5 be a stable tertiary cohomology operation associated with the above relation. We assume that (ϕ4, ϕ5) and ψ5 are chosen to be spin trivial in the sense of Theorem 3.7 of [14].Let ϕ0,0, ϕ1,1 be the stable Adams basic secondary cohomology operations associated with the relations:respectively.

scholarly journals Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence

2011 ◽  
Vol 7 (2) ◽  
pp. 203-347 ◽  
Author(s):  
Hans-Joachim Baues ◽  
Mamuka Jibladze
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Explicit Formulae ◽  
Cohomology Operations

AbstractWe describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulæ for the computation of the E3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.

Polynomials and the mod 2 Steenrod Algebra

2017 ◽  
Author(s):  
Grant Walker ◽  
Reginald M. W. Wood
Keyword(s):  
Steenrod Algebra

Polynomials and the mod 2 Steenrod Algebra

2017 ◽  
Author(s):  
Grant Walker ◽  
Reginald M. W. Wood
Keyword(s):  
Steenrod Algebra

A Note on Parabolic Subgroups and the Steenrod Algebra Action

2008 ◽  
Vol 15 (04) ◽  
pp. 689-698
Author(s):  
Nondas E. Kechagias
Keyword(s):  
Steenrod Algebra ◽  
Parabolic Subgroups ◽  
Generating Set ◽  
Modular Invariants ◽  
Dickson Algebra

The ring of modular invariants of parabolic subgroups has been described by Kuhn and Mitchell using Dickson algebra generators. We provide a new generating set which is closed under the Steenrod algebra action along with the relations between these elements.

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