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

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

This is also called Wu formula.

I also obtain 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{align}

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

I am looking for a generalization of (eq.1), such that

  • the $x_j \in H^j(M^d;\mathbb Z_2)$ is replaced by a more general

    $$ X_j \in H^j(M^d;\mathbb Z_{m}) $$

  • the Steenrod square $Sq$ is replaced by

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

is the Bockstein homomorphism associated to the extension $$\mathbb Z_n\stackrel{\cdot m}{\to}\mathbb Z_{nm}\to\mathbb Z_m$$ where $\cdot m$ is the group homomorphism given by multiplication by $m$. In particular, $$\beta_{(2,2^n)}=\frac{1}{2^n}\delta\mod2.$$

Question 1: What will be a generalization of (eq.1), when we replace the $$Sq^{1}=\beta_{(2,2)} \text{ to } \beta_{(n,m)},$$ 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: More specifically, what will be a generalization of (eq.1), when we $$ \text{ replace the } Sq^{1}=\beta_{(2,2)} \text{ to } \beta_{(2,4)},$$ and $$\text{ replace the } x_2 \in H^2(M^d;\mathbb Z_2) \text{ to } X_2 \in H^2(M^d;\mathbb Z_{4})?$$ Could we simply the relations (like (eq.1) formula between Wu class and the Steenrod square $$ \beta_{(2,4)}(X_2)=? $$ or $$ \beta_{(2,4)}(X_2 \cup X_j \cup X_{k})=? $$ where $X_j \in H^j(M^d;\mathbb Z_{4})$, similarly for $X_k$.


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