## 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 $$X_k$$,