I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces $X$, $Y$ (CW-complexes, say).
So far I had the following idea:
If two maps $f,g:X\rightarrow Y$ are homotopic, then they induce the same maps on all homotopy groups. Maybe there's a class of spaces for which the converse is also true. Then I would have a characterization via Homomorphisms between homotopy groups.
Does anybody know if there is a class of spaces for which this is true?
Best Answer
How about Eilenberg--Mac Lane spaces?
Let $G$ and $H$ be any groups. For pointed homotopy classes, $\langle K(G,1), K(H,1)\rangle $ is $\operatorname{Hom}(G,H)$, and for unpointed homotopy classes $[K(G,1), K(H,1)]$ is $\operatorname{Hom}(G,H)/H$, the orbits under conjugation by $H$.
When $n>1$ and $G$ and $H$ are abelian, we have $$\langle K(G,n),K(H,n)\rangle = [K(G,n), K(H,n)]=\operatorname{Hom}(G,H),$$ as pointed out by András in the comments.
Eilenberg--Mac Lane spaces also give counter-examples to the conjecture that maps $f,g:X\to Y$ between CW complexes are homotopic if they induce the same maps on homotopy groups. See my answer here.