# reference request – Postnikov square explicitly on a simplicial complex

$$DeclareMathOperatorZ{mathbb{Z}}$$

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?