Non-Naturality of the Splitting in the Universal Coefficient Formula I want to show the non-naturality of the splitting in the universal coefficient formula for homology. The s.e.s. is 
$$0\to H_q(X,X';R)\otimes_R N\to H_q(X,X';N)\to Tor^R_1(H_{q-1}(X,X';R),N)\to 0$$
where $R$ is a PID and $N$ an $R-$module. 
I found a counter example with the map $\mathbb{RP}^2\to S^2$ that collapses the 1-cell of $\mathbb{RP}^2$ to a point, but I don't get why this map works. 
 A: Look at $R=\mathbb Z, N = \mathbb Z/2$. 
Then you have the two short exact sequences $0\to H_2(\mathbb RP^2; \mathbb Z)\otimes \mathbb Z/2\to H_2(\mathbb RP^2;\mathbb Z/2)\to \mathrm{Tor}_1^\mathbb Z(H_1(\mathbb RP^2;\mathbb Z), \mathbb Z/2)\to 0$
and 
$0\to H_2(S^2; \mathbb Z)\otimes \mathbb Z/2\to H_2(S^2;\mathbb Z/2)\to \mathrm{Tor}_1^\mathbb Z(H_1(S^2;\mathbb Z), \mathbb Z/2)\to 0$
In the first exact sequence, the leftmost term is $0$, and in the second one it's the rightmost term which is $0$. 
If the splitting were natural, since $H_2(\mathbb RP^2; \mathbb Z)\otimes \mathbb Z/2\to H_2(S^2; \mathbb Z)\otimes \mathbb Z/2$ and $\mathrm{Tor}_1^\mathbb Z(H_1(\mathbb RP^2;\mathbb Z), \mathbb Z/2)\to \mathrm{Tor}_1^\mathbb Z(H_1(S^2;\mathbb Z), \mathbb Z/2)$ are both $0$ (because in each one, one of the terms is $0$), then $H_2(\mathbb RP^2;\mathbb Z/2)\to H_2(S^2;\mathbb Z/2)$ would be $0$ (can you see how this relates to naturality of the splitting ?)
But it's not $0$ (I don't know what's the easiest way to see this, here are at least two possibilities : 
a) Use Poincaré duality and try to convince yourself that your map sends the fundamental class (mod $2$) $[\mathbb RP^2]$ to $[S^2]$
b) Use the following cofiber sequence : $S^1\overset 2\to S^1\to \mathbb RP^2$ and note that it continues as $S^1\to S^1\to \mathbb RP^2\to S^2\to S^2$ and that the map $\mathbb RP^2\to S^2$ is the one you're looking for. This implies that the map $S^2\to S^2$ is $\Sigma 2 = 2 : S^2\to S^2$ so it induces $0$ in mod $2$ homology, and so the long exact sequence in homology of the cofiber sequence $\mathbb RP^2\to S^2\to S^2$ shows that the first map is an iso on $H_2(-;\mathbb Z/2)$   )
