# 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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