Let $h:S^1\to X$ be a continuous map. Show if $h$ is homotopic to a point, then $h$ can be extended to a continuous map $H:D^2\to X$.
Intuitively is a little bit obvious, let $F:S^1\times I\to X$ a homotopy, where $F(x,0)=h(x)$ and $F(x,1)=x_0 \in X$ because you can pass $h$ to the quotient, if $q:S^1\times I\to S^1\times I/S^1\times \{1\}$ is the quotient map so there is a continuous map $\bar h:S^1\times I/S^1\times \{1\}\to X$, such that $h=\bar h\circ q$.
There are also homeomorphisms between $D^2$ and $S^1\times I/S^1\times \{1\}$ and $S^1\times \{0\}$ and $S^1$. We know also that $F$ restricted to $S^1\times \{0\}$ is $f(x)$. Even with these informations I can't prove the question, I think we have to compose these maps to have a map $H$, but I can't prove that the restriction of $H$ in $S^1$ is $f$. The question is more subtle than I thought at the first sight.
I need help
Thanks