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Suppose $G$ is an elementary abelian $p$-group of rank n (for simplicity we can assume n=1). Denote by $\beta$ the well-known Bockstein boundary map from $H^1(G,\mathbb F_p)$ to $H^2(G,\mathbb F_p)$. I am looking for an explicit formula for $\beta(f)$ on $[g|h]$ if we know the value of $f$ on $G$.

Xingting
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    What is $g|h$ ? Do you mean $[g|h]$ from bar resolution ? – Demin Hu Apr 07 '13 at 22:23
  • yes, it means $[g|h]$ from bar resolution. – Xingting Apr 08 '13 at 07:17
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    I guess there's no other explicit formula appart from what can be deduced from the 'explicit' construction of the connection homomorphism in a cohomology long exact sequence comming from a short exact sequence of complexes. – Fernando Muro Apr 08 '13 at 07:29
  • I agree with Fernando. In particular, the first step of the construction requires you to lift a cocycle $f: P_1\to\mathbb{Z}/p$ to a cochain $\tilde f : P_1\to \mathbb{Z}/p^2$, and this will generally involve choices. That's why you end up with a cohomology class and not a cocycle. – Mark Grant Apr 08 '13 at 09:59
  • @ Fernando: If you fix a projective resolution (like the bar resolution chosen by the OP) you can be completely explicit as is shown in my answer. – Demin Hu Apr 08 '13 at 18:59

1 Answers1

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Write $G=\langle \sigma\rangle\cong \mathbb{Z}/p$. $f\in H^1(G,\mathbb{F}_p)$ can be taken as group homomorphism $f: G \to \mathbb{F}_p$. If $B$ denotes the bar resolution then $\beta(f)$ is represented by the cocycle $$B_2 \to \mathbb{F}_p,\;\; [\sigma^i,\sigma^j] \mapsto \begin{cases}f(\sigma) & , & i+j\ge p \newline 0 &,& i+j < p\end{cases}\qquad (0\le i,j < p)$$

For, $\beta(f)$ is the composition of the connecting homomorphism $\delta: H^1(G,\mathbb{F}_p) \to H^2(G,\mathbb{Z})$ that results from the short exact sequence $0 \to \mathbb{Z} \xrightarrow{\cdot p}\mathbb{Z}\to \mathbb{F}_p\to 0$ and the mod-p reduction $\rho: H^2(G,\mathbb{Z}) \to H^2(G,\mathbb{F}_p)$.

Let $f(\sigma)=k \mod p$. $\tilde{f}: B_1 \to \mathbb{Z},\;[\sigma^i]\mapsto ik\;(0\le i< p)$ is a lift of $f$ and the homomorphism $B_2 \xrightarrow{\partial}B_1 \xrightarrow{\tilde{f}}\mathbb{Z}$ is $$[\sigma^i,\sigma^j] \mapsto \begin{cases}pk & , & i+j \ge p \newline 0 &,& i+j< p\end{cases}\qquad (0\le i,j < p)$$ Since $\delta(f)$ is represented by the cocycle $\frac{1}{p}(\tilde{f}\circ \partial)$, the explicit formula above follows.

Demin Hu
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