I am working on Hatcher's problem 1.1.5.
Show that for a space $X$, the following three conditions are equivalent.
$\textit{a)}$ Every map $S^1\to X$ is homotopic to a constant map.
$\textit{b})$ Every map $S^1\to X$ extends to a map $D^2\to X$
$\textit {c})$ $\pi_1(X,x_0)=0$.
I have shown that $a$ implies $b$.
Now I want to prove that $b$ implies $c$. Therefore, I let $[f]\in\pi_1(X,x_0)$. Then, $f:I\to X$ with $f(0)=f(1)=x_0$. I want to show that $f$ is homotopic to the constant path. Since $S^1\cong I/_{0\sim 1}$, $f$ induces a map $f':S^1\to X$ with $f'(1)=x_0$. By $b$, this map can be extended to a map $\tilde{f'}:D^2\to X$. Since $D^2$ is convex, $D^2$ is contractible, there exists a homotopy $\tilde{H}:I\times D^2\to X$, $\tilde{H}_0=const_{x_0}$, $\tilde{H}_1=Id_{D^2}$. I don't know how I can go on from this, There is also a point where I have to transfer back to $f$ without $'$.
For $c$ implies $a$: Let $[f]\in\pi_1(X,x_0)$. Then $f:I\to X$ is homotopic to the constant path $const_{x_0}$. Also here, I don't know how to go on.
Hint: If $f':S^1 \to X$ extends to a map $\tilde f: D^2 \to X$, this means $f'$ can be written as a composition $$ (S^1,1) \xrightarrow i (D^2,1) \xrightarrow {\tilde f} (X,x_0) $$ where $i$ is the inclusion. Can you find a homotopy $H_t$ rel $1$ from $i$ to some constant map $c$? What if you compose it with $\tilde f$?
For (c) $\implies$ (a), given a map $g:S^1\to X$, can you find a homotopy from $g$ to a loop at $x_0$?