Stably, phantom maps (nonzero maps which are zero on homotopy) exist, but it’s not known if they exist between finite complexes (Freyd’s Generating Hypothesis). Unstably, it’s easy to find maps which are the same on homotopy but not homotopic (even between finite complexes), however I don’t know an example of a map which is the identity on homotopy but not homotopic to the identity.
I presume I could take $ \Omega^\infty(1+f)$ where $ f$ is a known stable phantom map — and I would be interested to see the details worked out — but known stable phantom maps seem to hinge crucially on $ \varprojlim^1$ issues, and so are “inherently infinitary”. I’d like to see an unstable example which is not “inherently infinitary”; concretely it would be nice to see an example on a finite complex, but I’m not wedded to this interpretation of “not inherently infinitary”. To sum up:
Question: What is an example of a self-map $ f: X \to X$ of a (pointed, connected) CW complex which is not homotopic to the identity but such that $ \pi_\ast(f): \pi_\ast(X) \to \pi_\ast(X)$ is the identity? Bonus points if $ X$ is a finite complex.
An equivalent question is: what’s an example of a space $ X$ such that $ Aut(X)$ doesn’t act faithfully on $ \pi_\ast(X)$ , where $ Aut(X)$ is the group of homotopy classes of self-homotopy-equivalences of $ X$ ?
Another way of putting this is: Whitehead’s theorem tells us that the functor $ \pi_\ast$ reflects isomorphisms, but does it reflect identities, i.e. is it faithful with respect to isomorphisms?
For that matter, I’m having trouble coming up with a space $ X$ such that $ Aut(X)$ doesn’t act faithfully on its homology, either.
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