I have some questions regarding Bockstein homomorphism in John McCleary's book (pg 455-456).
Q1) Is there a typo, is it supposed to be $\bar{u}\in H_n(X;\mathbb{Z}/r\mathbb{Z})$?
Q2) How do we see that $\partial(c)\neq 0$?
Q3) How do we conclude that $\partial(c)=rv$?
Q4) How does the boundary homomorphism $\partial$ actually look like explicitly in this case, and how do we know it takes $\bar{u}$ to $\{v\}\in H_{n-1}(X)$?
Thank you for any help.
Q1) Yes, it looks like a typo.
Q2) If $\partial c=0$, then $c \otimes 1$ would be in the image of $\text{red}_r$ and so the homology class $\overline{u}$ would go to zero under $\partial$.
Q3) The boundary map in $C_*(X) \otimes \mathbb{Z}/r$ is $\partial \otimes 1$. So $0=\partial (c \otimes 1) = (\partial c) \otimes 1$ means that $\partial c$ goes to zero when you reduce mod $r$, which means that it is in the image of multiplication by $r$.
Q4) The whole point of McCleary's discussion is to describe $\partial$ in this case. That is, he is taking the general description of the boundary map in homology arising from a short exact sequence of chain complexes, and he is analyzing what happens here. So I think this passage from the book already answers your question.