I am reading Hatcher's topology book and I have a question regarding the universal coefficient theorem page $264$, which is
I would like to know why this succession is split but not naturally, what is a counter example to show that this isomorphism is not natural? Thank you.
The fact that it splits is probably proved in Hatcher, during the proof of this theorem, so I will not comment on that unless you explicitly ask for that.
The fact that the splitting is nonnatural can be witnessed as follows : note that it suffices to find a chain map $f: C\to C'$ such that $H_n(C)\otimes G\to H_n(C')\otimes G$ and $\mathrm{Tor}(H_{n-1}(C),G)\to \mathrm{Tor}(H_{n-1}(C'),G)$ are zero but $H_n(C;G)\to H_n(C';G)$ is not (that's an exercise : prove that if the splitting were natural, then this situation could not happen).
An example of that is with the following complexes: $C=\mathbb{Z\xrightarrow{2} Z}$ and $C'= \mathbb{Z\xrightarrow{0} Z}$ and $f$ the chain map that is $0$ on degree $0$ and $1$ on degree $1$
Then $H_*(f) =0$ as an easy computation shows. Therefore the two maps we want to be $0$ are $0$.
However if we tensor these with $G=\mathbb{Z/2Z}$, we get $G\xrightarrow{0} G$ and $G\xrightarrow{0} G$ and $H_1(f; G) = (1 : G\to G)$.
In fact, we can generalize this argument at will, it will work with some minor adjustments whenever $G$ is not a flat module (of course when $G$ is flat the $\mathrm{Tor}$ terms vanish and so the splitting is natural)
