Talk:Steenrod algebra

Latest comment: 1 year ago by 77.8.16.45 in topic Coaction in generalized setting

Cohomological Operations

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"...Note that cohomology operations need not be group homomorphisms."

But in the definition of natural transformation, it says

"If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism ηX : F(X) → G(X) in D", 

so I deduce that cohomology operations need to be morphisms in the category of groups, i.e. group homomorphisms. Espigaymostaza (talk) 14:11, 10 January 2008 (UTC)Reply

yeah, it is a typo

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it is a type, you are right, the point is that it is in fact a morphism of gradd abelian groups, so if one forgets that cohomology is a ring then it is a morphism, but since it is just a collection of natural transformations, which may not be how you want to think about it pedagogically, it is really only a morphism from the abelian group H^n(X) ---> H^n+i(X). and i will fix this now, but what was meant was that it is not a morphism of rings, it is almost never a ring homomorphism, see the cartan fmla.

Sean, a student 06:47, 5 July 2008 (UTC)

Topological explanation

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I never really understood Steenrod squares until somebody told me this. Let   be the Eilenberg-MacLane_space, and note that degree m cohomology of X is classified by homotopy classes of maps into   that is,   Then the Steenrod squares are induced by composing with homotopy classes of maps  , so they are given by elements of   The Steenrod squares are just some subgroup of this. (If I remember correctly, they aren't the entire cohomology of E-M space, but a particularly easy subgroup to calculate. Note we restrict to dimensions where n \leq m.) The squares themselves are a basis for this subgroup, and the Adem relations can be calculated on E-M space, it follows they hold for all manifolds. Maybe someone else knows about this and can sat a little more, especially about why you pick this particular subgroup. 173.228.85.18 (talk) 12:45, 12 May 2011 (UTC)Reply

Examples Needed

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This page should discuss examples of the steenrod squaring operation. This should inlcude  . — Preceding unsigned comment added by 71.212.185.82 (talk) 02:00, 15 August 2017 (UTC)Reply

Construction + Citations

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  • Discuss McCleary, 4.4, 6.4, 8.3
  • Discuss stable cohomology operations
  • Relate to computing the cohomology of all mod p eilenberg-maclane spaces (given in Hatcher spectral sequences)
  • Discuss some of the computations of Adams spectral sequence coming from Steenrod squares (McCleary)

Coaction in generalized setting

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The coaction is jot induced by the product on E but by the unit from S to E. The latter would induce an action since the second variable of hom is covariant. 77.8.16.45 (talk) 17:36, 24 March 2023 (UTC)Reply