Bockstein connecting operation

Assume the Axiom of Choice from The Axiom of Choice. Let

$$0⟶A\stackrel{i}{\to }B\stackrel{q}{\to }C⟶0$$

be a short exact sequence of abelian groups. Applying singular cochains degree by degree gives a short exact sequence of cochain complexes: injectivity on the left follows from injectivity of $i$, and surjectivity on the right follows as follows. A cochain is a function on the set $S_n(X)$ of singular $n$-simplices by Singular cochain complex with coefficients. For a $C$-valued cochain $c$, AC is used exactly once to choose, simultaneously for every $s\in S_n(X)$, an element of the nonempty fibre $q^{-1}(c(s))$. The resulting function is a $B$-valued cochain $b$ with $q_{\ast }b=c$.

For a cocycle $c\in C^n(X;C)$ choose such a lift $b$. Since $q_{\ast }(\delta b)=\delta c=0$, exactness gives a unique $a\in C^{n+1}(X;A)$ satisfying $i_{\ast }a=\delta b$. The Bockstein connecting operation is

$$\beta \text{\hspace{0.17em}⁣}:H^n(X;C)⟶H^{n+1}(X;A),\beta [c]=[a]\mathrm{.}$$

This is the coefficient-sequence analogue of the cochain connector in Long exact sequence of a pair in singular cohomology. Its independence of $b$ and $c$ is proved in the next lemma.

For an integer $m\ge 1$, two Bocksteins must be distinguished:

• the integral Bockstein $\tilde{\beta }\text{\hspace{0.17em}⁣}:H^n(X;\mathbb{Z}\mathrm{/}m)\to H^{n+1}(X;\mathbb{Z})$ comes from $0\to \mathbb{Z}\stackrel{m}{\to }\mathbb{Z}\to \mathbb{Z}\mathrm{/}m\to 0$;
• the mod-$m$ Bockstein $\beta \text{\hspace{0.17em}⁣}:H^n(X;\mathbb{Z}\mathrm{/}m)\to H^{n+1}(X;\mathbb{Z}\mathrm{/}m)$ comes from $0\to \mathbb{Z}\mathrm{/}m\stackrel{m}{\to }\mathbb{Z}\mathrm{/}{m}^2\to \mathbb{Z}\mathrm{/}m\to 0$.

In these two cyclic sequences, least nonnegative residue representatives give the required cochain lift without AC. If $X$ is empty, or if $n<0$, the source cochain group is zero and the lift is unique. When $m=1$ both cyclic Bocksteins have zero source and hence are the zero operation. The mod-$m$ target is also zero, whereas the integral target $H^{n+1}(X;\mathbb{Z})$ need not vanish.