Talk:Steenrod algebra

Cohomological Operations

"...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

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

I never really understood Steenrod squares until somebody told me this. Let $K(\mathbb {Z} _{2},m)$ be the Eilenberg-MacLane_space, and note that degree m cohomology of X is classified by homotopy classes of maps into $K(\mathbb {Z} _{2},m),$ that is, $H^{m}(X)\cong [X,K(\mathbb {Z} _{2},n)].$ Then the Steenrod squares are induced by composing with homotopy classes of maps $K(\mathbb {Z} _{2},m)\to K(\mathbb {Z} _{2},m+n)$ , so they are given by elements of $H^{n+m}(K(\mathbb {Z} _{2},n)).$ 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

This page should discuss examples of the steenrod squaring operation. This should inlcude $\mathbb {RP} ^{\infty }$ . — Preceding unsigned comment added by 71.212.185.82 (talk) 02:00, 15 August 2017 (UTC)[reply]

Construction + Citations

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

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]