Let $A, B, C$ be finite Abelian groups fitting in a short exact sequence $$ 1 \rightarrow A\overset{\iota}{\rightarrow} B\overset{\pi}{\rightarrow} C\rightarrow 1 $$ This determines a class $[\epsilon]\in H^2(C,A)$ measuring the failure of the sequence to split: $$ s(c_1)+s(c_2)=s(c_1+c_2)+\iota(\epsilon (c_1,c_2)) $$ where $s:C\rightarrow A$ is any section of $\pi$ (different sections gives different representatives of $[\epsilon]$). By Pontryagin duality we have a dual sequence $$ 1 \rightarrow C^\vee\overset{\pi ^\vee}{\rightarrow} B^\vee\overset{\iota ^\vee}{\rightarrow} A^\vee\rightarrow 1 $$ and thus a dual class $\epsilon^\vee \in H^2(A^\vee,C^\vee)$.

I think this should be naturally defined explicitely in terms of $c$, but I don't see how. Part of my problem is that the only "explicit" equation in which $\epsilon$ enters is throught the section $s$, which however does not gives something natural under duality. How can I write down an explicit formula for $$ \epsilon^\vee(\alpha_1,\alpha_2) c \in \mathbb{R}/\mathbb{Z} \ , \ \ \ \ \ \alpha_1,\alpha_2 \in A^\vee \ , \ \ c\in C $$ ?

Let $A, B, C$ be finite Abelian groups fitting in a short exact sequence $$ 1 \rightarrow A\overset{\iota}{\rightarrow} B\overset{\pi}{\rightarrow} C\rightarrow 1 $$ This determines a class $[\epsilon]\in H^2(C,A)$ measuring the failure of the sequence to split: $$ s(c_1)+s(c_2)=s(c_1+c_2)+\iota(\epsilon (c_1,c_2)) $$ where $s:C\rightarrow A$ is any section of $\pi$ (different sections gives different representatives of $[\epsilon]$). By Pontryagin duality we have a dual sequence $$ 1 \rightarrow C^\vee\overset{\pi ^\vee}{\rightarrow} B^\vee\overset{\iota ^\vee}{\rightarrow} A^\vee\rightarrow 1 $$ and thus a dual class $\epsilon^\vee \in H^2(A^\vee,C^\vee)$.

I think this should be naturally defined explicitly in terms of $\epsilon$, but I don't see how. Part of my problem is that the only "explicit" equation in which $\epsilon$ enters is through the section $s$, which however does not gives something natural under duality. How can I write down an explicit formula for $$ \epsilon^\vee(\alpha_1,\alpha_2) c \in \mathbb{R}/\mathbb{Z} \ , \ \ \ \ \ \alpha_1,\alpha_2 \in A^\vee \ , \ \ c\in C $$ ?

Let $A, B, C$ be finite Abelian groups fitting in a short exact sequence $$ 1 \rightarrow A\overset{\iota}{\rightarrow} B\overset{\pi}{\rightarrow} C\rightarrow 1 $$ This determines a class $[\epsilon]\in H^2(C,A)$ measuring the failure of the sequence to split: $$ s(c_1)+s(c_2)=s(c_1+c_2)+\iota(\epsilon (c_1,c_2) $$ where $s:C\rightarrow A$ is any section of $\pi$ (different sections gives different representatives of $[\epsilon]$). By Pontryagin duality we have a dual sequence $$ 1 \rightarrow C^\vee\overset{\pi ^\vee}{\rightarrow} B^\vee\overset{\iota ^\vee}{\rightarrow} A^\vee\rightarrow 1 $$ and thus a dual class $\epsilon^\vee \in H^2(A^\vee,C^\vee)$.

I think this should be naturally defined explicitely in terms of $c$, but I don't see how. Part of my problem is that the only "explicit" equation in which $\epsilon$ enters is throught the section $s$, which however does not gives something natural under duality. How can I write down an explicit formula for $$ \epsilon(\alpha_1,\alpha_2) c \in \mathbb{R}/\mathbb{Z} \ , \ \ \ \ \ \alpha_1,\alpha_2 \in A^\vee \ , \ \ c\in C $$ ?

Let $A, B, C$ be finite Abelian groups fitting in a short exact sequence $$ 1 \rightarrow A\overset{\iota}{\rightarrow} B\overset{\pi}{\rightarrow} C\rightarrow 1 $$ This determines a class $[\epsilon]\in H^2(C,A)$ measuring the failure of the sequence to split: $$ s(c_1)+s(c_2)=s(c_1+c_2)+\iota(\epsilon (c_1,c_2)) $$ where $s:C\rightarrow A$ is any section of $\pi$ (different sections gives different representatives of $[\epsilon]$). By Pontryagin duality we have a dual sequence $$ 1 \rightarrow C^\vee\overset{\pi ^\vee}{\rightarrow} B^\vee\overset{\iota ^\vee}{\rightarrow} A^\vee\rightarrow 1 $$ and thus a dual class $\epsilon^\vee \in H^2(A^\vee,C^\vee)$.

I think this should be naturally defined explicitely in terms of $c$, but I don't see how. Part of my problem is that the only "explicit" equation in which $\epsilon$ enters is throught the section $s$, which however does not gives something natural under duality. How can I write down an explicit formula for $$ \epsilon^\vee(\alpha_1,\alpha_2) c \in \mathbb{R}/\mathbb{Z} \ , \ \ \ \ \ \alpha_1,\alpha_2 \in A^\vee \ , \ \ c\in C $$ ?