at.algebraic topology – Dual Steenrod squares

Attach the bottom ring $ mathbb {F} _2 $ and let it go $ X $ to be a space with finite homology. Then we have an isomorphism $ Phi ^ i_X: H_i (X) to H ^ i (X) ^ *, a mapsto langle-, a rangle $ So we can define the two Steenrod squares
$$ mathrm {Sq} _r: H_i (X) stackrel { Phi_X ^ i} { to} H ^ i (X) ^ * stackrel {( mathrm {Sq} ^ r) ^ *} { to} H ^ {ir} (X) ^ * stackrel {( Phi ^ {ir} _X) ^ {- 1}} { an} H_ {ir} (X). $$
I want to understand which formulas satisfy them. Many things seem simple: For example, the "dual" Cartan formula requires simplification $ Delta: H_i (X) an bigoplus_ {p + q = i} H_p (X) times H_q (X) $ that's twice the cup product, and you should get that
$$ Delta_ {i-r} circ mathrm {Sq} _r = sum_ {p + q = r} ( mathrm {Sq} _p otimes mathrm {Sq} _q) circ Delta_i. $$
What I'm trying to dualize now is
$$ mathrm {Sq} ^ nx = x smile x ~~~ text {for} | x | = n. $$
The right side is obviously the diagonally pre-assembled cup product $ nabla: H ^ n (X) to H ^ n (X) ^ { times 2} subseteq bigoplus_ {p + q = 2n} H ^ p (X) times H ^ q (X) $, We can find a card that is double to this diagonal and shape
$$ D: bigoplus_ {p + q = 2n} H_p (X) times H_q (X) to H_n (X), a times b mapsto a , # , b. $$
We should then decide $ a in H_ {2n} (X) $ the statement
$$ mathrm {Sq} _n (a) = (D circ Delta) (a). $$
That does not look very revealing. Is this "wrongly graded product"? $ D $ have any meaning? At least it fulfills the following property for all $ alpha in H ^ n (X) $
$$ langle alpha, a , # , b rangle = langle alpha ^ { times 2}, a times b rangle. $$

Reference request – What is the square of the dual Steenrod algebra?

The dual Steenrod algebra has generators $ xi_n $ and these have conjugates that are often tagged $ zeta_n $, I am curious about the left and right actions of the Steenrod algebra in their dual and especially on the entire square. I saw that in papers $ ( xi_n) Sq = xi_n + xi_ {n-1} $ and $ Sq ( xi_n) = xi_n + xi_ {n-1} ^ 2 $ [1], On the other side, I saw that $ ( zeta_n) Sq = zeta_n + zeta_ {n-1} ^ 2 + dots + zeta_1 ^ {2 ^ {n-1}} + 1 $ [2], I can not find a reference for the left total square anywhere $ zeta_n $, I am not sure how to prove these actions, though it seems to me that this should result from fairly elementary Kronecker product arithmetic and duality knowledge.

I am interested in either a reference for the left total square or a way to prove it.

[1] See for example Mahowald – bo resolutions, page 369.

[2] Bruner, May, McClure, Steinberger – $ H_ infty $ Ring Spectra and Their Applications, page 78. (There is a typo: 1 should be $ i $.)

at.algebraic topology – generalization of the formula between the Wu class and the Steenrod square

I know that on the tangent bundle of $ M ^ d $satisfy the corresponding Wu class and the Steenrod square

$$
Sq ^ {d-j} (x_j) = u_ {d-j} x_j, text {for any} x_j in H ^ j (M ^ d; mathbb Z_2).
tag {eq.1} $$

This is also known as Wu formula.

I also get the Wu class as
begin {align}
u_0 & = 1,

u_1 = w_1,
\
u_2 = w_1 ^ 2 + w_2,
\
u_3 & = w_1w_2,

u_4 = w_1 ^ 4 + w_2 ^ 2 + w_1w_3 + w_4,
\
u_5 & = w_1 ^ 3w_2 + w_1w_2 ^ 2 + w_1 ^ 2w_3 + w_1w_4.
end

I also know the related Steenrod relationship
$$
Sq ^ n (xy) = sum_ {i = 0} ^ n Sq ^ i (x) Sq ^ {n-i} (y).
$$

I am looking for a generalization of (equation 1), so that

$$ beta _ {(n, m)}: H ^ * (-, mathbb Z_ {m}) to H ^ {* + 1} (-, mathbb Z_ {n}) $$

is the Bockstein homomorphism associated with the extension
$$ mathbb Z_n stackrel { cdot m} { to} mathbb Z_ {nm} an mathbb Z_m $$ from where $ cdot m $ is the group homomorphism given by multiplication $ m $, Particularly, $$ beta _ {(2,2 ^ n)} = frac {1} {2 ^ n} delta mod2. $$

Question 1: What will become generalization from (Eq.1) if we replace that
$$ Sq ^ {1} = beta _ {(2,2)} text {
to
beta _ {(n, m)}, $$

and replace the
$$ x_j in H ^ j (M ^ d; mathbb Z_2) text {
to
X_j in H ^ j (M ^ d; mathbb Z_ {m})? $$

Question 2: Specifically, what will happen generalization from (Eq.1) if we
$$ text {replaces} Sq ^ {1} = beta _ {(2,2)} text {
to
beta _ {(2,4)}, $$

and
$$ text {replaces the} x_2 in H ^ 2 (M ^ d; mathbb Z_2) text {
to
X_2 in H2 (M ^ d; mathbb Z_ {4})? $$

Could we just use the relation (like (Equation 1)) between the Wu class and the Steenrod square
$$
beta _ {(2,4)} (X_2) =?
$$

or
$$
beta _ {(2,4)} (X_2 cup X_j cup X_ {k}) =?
$$

from where $ X_j in H ^ j (M ^ d; mathbb Z_ {4}) $similar for $ X_k $,