Given 2 $CW$-complexes $X,Y$ and maps $$f,g:X \rightarrow Y$$ such that both $f_*: \pi_i(X) \rightarrow \pi_i(Y)$ and $f_*: \pi_i(X) \rightarrow \pi_i(Y)$ are isomorphisms $\forall ~i \geq 0$
Is it true that $f$ and $g$ are homotopic maps?
I do know that if we just have $f_* \cong g_*$ then $f$ and $g$ need not be homotopic but all counter examples seems to be maps that induce the 0 map on the homotopy groups but are not homotopic to the constant map. Do we have such counterexamples when $f_*$ and $g_*$ are isomorphisms as well?
Best Answer
There are still counterexamples when $f_*$ and $g_*$ are isomorphisms, and indeed when they are the identity. Shih shows in Corollary 2 that, if $X$ is a simply connected space with two nontrivial homotopy groups $A=\pi_n X$ and $B=\pi_m X$, then the group of homotopy classes of self-homotopy equivalences of $X$ inducing the identity on $A$ and $B$ is the cohomology group $H^m(K(A,n);B)$.