Bockstein homomorphism

Remarks

The same construction on chains gives a homology Bockstein $H_n(X; C) \to H_{n-1}(X; A)$ in homology with coefficients. The most used case is $0 \to \Z/p \to \Z/p^2 \to \Z/p \to 0$, which gives a natural map $\beta \colon H^n(X; \F_p) \to H^{n+1}(X; \F_p)$ with $\beta \circ \beta = 0$.

Example: let $X = \R P^2$ and $p = 2$, so $H^1(X; \Z/2) \cong H^2(X; \Z/2) \cong \Z/2$, while $H^1(X; \Z/4) \cong \Z/2$ and $H^2(X; \Z/4) \cong \Z/2$. The map $H^1(X; \Z/4) \to H^1(X; \Z/2)$ is zero, since the class of order two in $H^1(X; \Z/4)$ reduces to zero mod $2$. So by exactness the generator of $H^1(X; \Z/2)$ has nonzero Bockstein, which is the generator of $H^2(X; \Z/2)$.

Non-example of a nonzero Bockstein: for the split sequence $0 \to \Z/2 \to \Z/2 \oplus \Z/2 \to \Z/2 \to 0$, every class lifts, and the Bockstein is the zero map. In general $\beta = 0$ whenever the coefficient sequence splits.

For $p = 2$ the Bockstein on mod $2$ cohomology equals the Steenrod square $Sq^1$. For the sequence $0 \to \Z \xrightarrow{p} \Z \to \Z/p \to 0$ one gets the integral Bockstein $H^n(X; \Z/p) \to H^{n+1}(X; \Z)$, whose image consists of $p$-torsion classes.