Hello pricey customer to our community We will proffer you an answer to this query reference request – Postnikov sq. explicitly on a simplicial difficult ,and the retort will breathe typical via documented info sources, We welcome you and proffer you contemporary questions and solutions, Many customer are questioning concerning the retort to this query.

## 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?

we’ll proffer you the answer to reference request – Postnikov sq. explicitly on a simplicial difficult query through our community which brings all of the solutions from a number of reliable sources.

## Add comment