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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 $\mathbb{Z}/2$.

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# Examples for non-naturality of universal coefficients theorem

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?