$DeclareMathOperatorZ{mathbb{Z}}$

Following Wikipedia, a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by Postnikov (1949). Eilenberg (1952) described a generalization taking classes in $H^t$ to $H^{2t+1}$ which requires that $t$ is odd.

Here I consider the specific Postnikov square $mathfrak{P}_3$ (hopefully) defined by $$mathfrak{P}_3: H^2(-,Z_{3^k})to H^5(-,Z_{3^{k+1}})$$ given by

$$

mathfrak{P}_3(u)=beta_{(3^{k+1},3^k)}(ucup u)

$$

where $beta_{(3^{k+1},3^k)}$ is the Bockstein homomorphism associated with $0toZ_{3^{k+1}}toZ_{3^{2k+1}}toZ_{3^k}to0$ and $u in H^2(M,Z_{3^k})$.

Let us focus on $k=1$ case,

$$mathfrak{P}_3(u)=beta_{(3^{2},3)}(ucup u)=beta_{(9,3)}(ucup u).$$

Here $u in H^2(M,Z_3)$ that we can define as a 2nd cohomology class (also 2-cocycle) with a $Z_3$ coefficient on a manifold $M$. For example, let us take $u$ to be on a 2-simplex with 3 vertices $(0-1-2)$, then we denote the data $u$ assign on this 2-simplex as:

$$

u_{(0-1-2)}.

$$

## Question

Then How do we write $mathfrak{P}_3(u)=beta_{(3^{2},3)}(ucup u)=beta_{(9,3)}(ucup u)$ on

a 5-simplex with 6 vertices $(0-1-2-3-4-5)$. Say, we start with the cup product $ucup u$ that can be defined on a 4-simplex with 5 vertices $(0-1-2-3-4)$, thus

$$

(ucup u)_{(0-1-2-3-4)} = u_{(0-1-2)} u_{(2-3-4)}.

$$

which is a product of two 2-cocycles on the 2-simplex with 3 vertices $(0-1-2)$ and another 2-simplex with 3 vertices $(2-3-4)$. How do we write explicitly on the 5-simplex $(0-1-2-3-4-5)$:

$$

mathfrak{P}_3(u)_{(0-1-2-3-4-5)}=beta_{(9,3)}(u_{(i-j-k)} u_{(k-l-m)})=?

$$

So we have $mathfrak{P}_3: H^2(-,Z_{3})to H^5(-,Z_{9})$ on the 5-simplex?