Does anyone have good examples of a space $X$ and a map $f: X \to X$ so that $f\_\*: H\_\*(X) \to H\_\*(X)$ is the identity but (e.g.) $f\_\*: H\_\*(X; \mathbb{F}\_2) \to H\_\*(X; \mathbb{F}\_2)$ is not the identity?


**Edit:** As mentioned in the comments, $f\_\*$ is an isomorphism on the mod-2 homology, but I don't see why it needs to be the identity. More precisely, the exact sequence
$$
C(X; \mathbb{Z}) \overset{\times 2}{\longrightarrow} C(X; \mathbb{Z}) \longrightarrow C(X;\mathbb{Z}/2)
$$
gives an exact triangle in homology, which in turn induces a 2-step filtration on $H\_\*(X; \mathbb{Z}/2)$ (where one step is the image of the map $H\_\*(X;\mathbb{Z}) \to H\_\*(X;\mathbb{Z}/2)$). The assumption that $f\_\*$ is the identity on integral homology implies that it is the identity on the associated graded space to this filtration, but that still doesn't imply it is the identity.


I came across a similar phenomenon in the context of Heegaard Floer homology, with the rings $(\mathbb{Z}/2)[U]$ and $\mathbb{Z}/2$ playing the roles of $\mathbb{Z}$ and $\mathbb{Z}/2$.



Examples for non-naturality of universal coefficients theorem
To expand on my comment, let $M = \mathbf{R}P^2$ be the Moore space with (reduced) homology concentrated in dimension $1$. Let $f:M \to \Sigma M$ be the map
$$ M \to S^2 \to \Sigma M $$
given by collapsing the $1$-skeleton of $M$ and then including the bottom cell into $\Sigma M$. This map induces $0$ on $\tilde H\_\ast({-};\mathbf{Z})$ for dimension reasons. However, the map $f$ is an isomorphism on $H\_2({-};\mathbf{Z}/2)$; this follows from the long exact sequences
$$ \dotsb \to H\_2(S^1;\mathbf{Z}/2) \to H\_2(M;\mathbf{Z}/2) \to H\_2(S^2;\mathbf{Z}/2) \to \dotsb $$
and
$$ \dotsb \to H\_2(S^2;\mathbf{Z}/2) \to H\_2(\Sigma M;\mathbf{Z}/2) \to H\_2(S^3;\mathbf{Z}/2) \to \dotsb .$$
This is an example of "non-naturality."


To obtain a self-map of a space $X$ that is the identity on $H\_\ast(X;\mathbf{Z})$ but not on $H\_\ast(X;\mathbf{Z}/2)$, we follow Tyler's suggestion: let $X = \Sigma M \vee \Sigma^2 M$. Since $X$ is a co-$H$-space, we can add maps in $[X,X]$. Let $g:X\to X$ be the sum of $1\_X$ and the map
$$ X \to \Sigma M \xrightarrow{\Sigma f} \Sigma^2 M \to X $$
where the first map collapses $\Sigma^2 M$ and the third map is the inclusion of $\Sigma^2 M$. The induced map in homology is $1 + \Sigma f\_\ast$. In $H\_\ast(X;\mathbf{Z})$, this is $1$. However, in $H\_3(X;\mathbf{Z}/2)$, the map $g$ is not the identity, since $\Sigma f\_\ast$ is nonzero. If we fix the basis of $H\_3(X;\mathbf{Z}/2)$ given by the wedge decomposition of $X$, then $g\_\ast$ is represented by the matrix


$$ \begin{pmatrix} 1 & 1 \\\\ 0 & 1 \end{pmatrix}. $$


