Let $f,g:X\longrightarrow X$ be two continuous maps. Recall that $f$ and $g$ are homotopic if there exists a continuous map $F:X\times I\longrightarrow X$ so that $F(x,0)=f(x)$ and $F(x,1)=g(x)$.
Is there a known result (in connection to homotopy or homology groups) from which we can obtain $f\simeq g$?
The homotopy classification of maps $X \to Y$ is clearly related to that of finding the path components of the function space $Y^X$.
However for some practical answers related to CW-complexes and cohomology I refer you to
Ellis, G.J. Homotopy classification the J.H.C. Whitehead way. Exposition. Math. 6~(2) (1988) 97--110.
A pdf is available here.