TITLE: Pontryagin square, Postnikov square and their consistency formulas
QUESTION [5 upvotes]: $\mathcal{P}_2$ is Pontryagin square
$$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$

$\mathfrak{P}$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$




question (i) Can Pontryagin square and Postnikov square coincide with the Steenrod square, or the Generalized "Bockstein homomorphism" $\beta_p$, $\beta_p'$, $\beta_{2^n}$ or other homomorphisms?

question (ii) Are there useful consistency formulas for these above "Pontryagin square" and "Postnikov square"?




Comments about question (i)
We know for $\mathbb Z_2$-valued cocycles $z_n$,
$$
Sq^{n-k}(z_n) \equiv z_n\cup_{k} z_n
$$
is always a cocycle.  Here $Sq$ is called the Steenrod square.
More generally
$h_n \cup_{k} h_n$ is a cocycle if $n+k =$ odd and $h_n$ is a cocycle.
If we define a generalized Steenrod square for
cochains $c_n$:
$$
\tilde Sq^{n-k} c_n \equiv c_n\cup_{k} c_n +  c_n\cup_{k+1} d c_n .
$$
We can check
$$
 d \tilde  Sq^{k} c_n = d(
c_n\cup_{n-k} c_n +  c_n\cup_{n-k+1} d c_n )
$$
$$
= \tilde  Sq^k d c_n, \;\;\;\;  k=\text{odd}
$$
$$
=\tilde  Sq^k d c_n +(-)^{n} 2 \tilde   Sq^{k+1} c_n ,  \;\;\;\; k=\text{odd}.
$$
This $$\tilde  Sq^{2} c_2 \equiv c_2\cup_{0} c_2 +  c_2\cup_{1} d c_2$$ almost is the same as the Pontryagin square $\mathcal{P}_2$ above, for ($i=1,k=2$ above)
$$H^{2}(M,\mathbb Z_{2})\to H^{4}(M,\mathbb{Z}_{4}),$$
for
$$\mathcal{P}_2 (x_2) \equiv x_2\cup_{0} x_2 +  x_2\cup_{1} d x_2.$$

Are these generalized Steenrod squares known? ($\tilde  Sq^{n-k} c_n$) Where can I find more discussions along this?


Comments about question (ii)
For example, for Steenrod square, the total Stiefel-Whitney class $w=1+w_1+w_2+\cdots$ is related to the
total Wu class $u=1+u_1+u_2+\cdots$ through the total Steenrod square
$$ w=Sq(u),\ \ \ Sq=1+Sq^1+Sq^2+ \cdots .
$$
Therefore,
$w_n=\sum_{i=0}^n Sq^i (u_{n-i})$.
The Steenrod squares have the following properties:
$$
Sq^i(x_j) =0, \  i>j, \ \ 
Sq^j(x_j) =x_jx_j,  \ \  Sq^0=1,
$$

Do we have something similar for thse "Bockstein homomorphism?" $\beta_p$, $\beta_p'$, $\beta_{2^n}$?

REPLY [7 votes]: You may be interested in the following paper:
Massey, W. S., Pontryagin squares in the Thom space of a bundle, Pac. J. Math. 31, 133-142 (1969). ZBL0188.28504.
Massey proves an analogue for the Pontryagin square of Thom's formula $w_k=\Phi^{-1}Sq^k(u)$ for the Stiefel-Whitney classes, which seems relevant to your question (ii).