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reference request – Postnikov sq. explicitly on a simplicial difficult retort

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reference request – Postnikov sq. explicitly on a simplicial difficult

$DeclareMathOperatorZ{mathbb{Z}}$

Following Wikipedia, a Postnikov sq. is a inescapable cohomology operation from a primary cohomology group $H^1$ to a 3rd cohomology group $H^3$, launched by Postnikov (1949). Eilenberg (1952) described a generalization taking lessons in $H^t$ to $H^{2t+1}$ which requires that $t$ is weird.

Here I assume the particular Postnikov sq. $mathfrak{P}_3$ (hopefully) outlined by $$mathfrak{P}_3: H^2(-,Z_{3^okay})to H^5(-,Z_{3^{okay+1}})$$ given by
$$
mathfrak{P}_3(u)=beta_{(3^{okay+1},3^okay)}(ucup u)
$$

the place $beta_{(3^{okay+1},3^okay)}$ is the Bockstein homomorphism related to $0toZ_{3^{okay+1}}toZ_{3^{2k+1}}toZ_{3^okay}to0$ and $u in H^2(M,Z_{3^okay})$.

Let us give attention to $okay=1$ illustration,
$$mathfrak{P}_3(u)=beta_{(3^{2},3)}(ucup u)=beta_{(9,3)}(ucup u).$$
Here $u in H^2(M,Z_3)$ that we are able to outline as a 2nd cohomology class (too 2-cocycle) with a $Z_3$ coefficient on a manifold $M$. For occasion, allow us to maintain $u$ to breathe on a 2-simplex with 3 vertices $(0-1-2)$, then we denote the information $u$ allot on this 2-simplex as:
$$
u_{(0-1-2)}.
$$

Question

Then How assassinate 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 commence with the cup product $ucup u$ that may breathe outlined 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 one other 2-simplex with 3 vertices $(2-3-4)$. How assassinate 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 maintain $mathfrak{P}_3: H^2(-,Z_{3})to H^5(-,Z_{9})$ on the 5-simplex?

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