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)

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)

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)

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)