# 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. – Lee Mosher 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) – berndt 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. – berndt 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 – user31480 Feb 27 '19 at 15:45
• @MoisheKohan In fact, this theorem is false. The original reference seems to be from Shih, in 1964: projecteuclid.org/download/pdf_1/euclid.bams/1183526014 – Kevin Arlin Jan 19 at 23:57

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.