Let $f:X\to Y$ be a map between top spaces $X$ and $Y$. Is the following true?
If $X$ is contractible then $f$ must be nullhomotopic.
Here is an argument:
Since $X$ is contractile then $id_X\simeq c$ where $c:X\to X;x\mapsto *_X$. Let $H_t:X\to X$ be this homotopy, that is, $H_0=id_X$ and $H_1=c$. then the map $G_t:X\to Y; x\mapsto f\circ H_t(x)$ is a homotopy between $f$ and $c':X\to Y; x\to *_Y$.
I don't see what is wrong with this argument since it can't be true knowing that $I$ is contractible hence all loops are nullhomotopic which implies that all fundamental groups are trivial!!