# If the map induces identity on all homotopic groups then it is homotopic to identity

Recall that, in general, maps of CW complexes $X\to Y$ which induce the same maps of all homotopy groups $\pi_*(X,x)\to \pi_*(Y,y)$, need not be homotopic.

Assume however, that we have a continuous self-map $f: (X,x) \rightarrow (X,x)$ of a connected CW complex $X$ which induces identity morphisms on all homotopy groups. Is it true that $f$ is actually homotopic to the identity map on $X$?

Note that in this setting Whitehead's theorem says $f$ is a homotopy equivalence.

• This is a standard theorem whose proof you can look up in any algebraic topology book. I recommend Hatcher's book. Unless you have specific questions about understanding such a proof, I'm voting to close. Feb 1 '17 at 21:37
• In Hatcher there is only a proof of Whitehead theorem " weak equivalence implies homotopy equivalence" and if you read carefully you will see that the question is not about the proof of Whitehead theorem. I understand that probably my question very easily follows from W-ed Theorem. if this is the case, can you please explain how it actually follows. As from what I see I think you read the question as " How to prove that weak equivalence implies homotopy equivalence". 2) Also if it happens that I am wrong could you please tell me in which chapter of Hatcher exactly this is proven? ( Not Wh-ed) Feb 1 '17 at 22:09
• Moishe Cohen, thank you, my doubt was if one should actually reprove the Whithead or this sort of uniqueness is a purely logical consequence of Whitehead. I looked not only in Hatcher, but in other well known algebraic topology books ( such as May) and nowhere found this sort of " refined Whitehead"- 1-1 correspondense between induced morphisms on homotopy groups and homotopy classes of maps. While the proof is analogous to usual Whithead, it seems little strange to me that nobody pays attention. Feb 2 '17 at 9:30
• Maybe this paper is of interest: SELF HOMOTOPY EQUIVALENCES WHICH INDUCE THE IDENTITY ON HOMOLOGY, COHOMOLOGY OR HOMOTOPY GROUPS, by Martin ARKOWITZ and Ken-ichi MARUYAMA sciencedirect.com/science/article/pii/S0166864197001624 Feb 27 '19 at 15:45
• Thanks. Now that I think about it, might there be an extra condition, namely that the homomorphisms on the higher homotopy groups are $\pi_1$-equivariant, or something like that? Jan 20 '20 at 0:54

Shih proved that, if $$X$$ is simply connected with two nontrivial homotopy groups, then the group of self-homotopy equivalences of $$X$$ inducing the identity on the homotopy groups is naturally identified the cohomology group $$H^m(K(\pi_n(X),n),\pi_m(X))$$, and so is frequently nontrivial.