The following are equivalent for a topological space X according to a problem in Hatcher.
$1$)Every continuous map $S^1 \to X$ is homotopic to a constant map.
$2$)Every continuous map $S^1 \to X$ extends to a continuous map $D^2 \to X$.
$3$)$\pi_1(X,x_0)=0$ for every $x_0 \in X$.
I am examining the proof of $2$) implies $3$).
Proof- Let $h:S^1 \to X$, then there is an extension $k:D^2 \to X$. If we let $j$ be the inclusion $S^1 \to D^2$, then $h=k \circ j$ and $h_*=k_* \circ j_*$. Since $\pi_1(D^2,s_0)=0$ for each $s_0 \in S^1$, then h* sends each element of $\pi_1(S^1,s_0)$ to the identity in $\pi_1(X,h(s_0))$.
So I have two issues. First, this only shows $h_*$ is the trivial homomorphism not $\pi_1(X,h(s_0))=0$. Second, even if I fix that then I seem to have only proved it for $x_0$ in the image of $h$. So if someone could help me with this I would be grateful.
Best Answer
Hint: A nullhomotopy $F:\mathbb{S}^1\times [0,1]\to X$ is equivalent to a map $G:\mathbb{D}^2\to X$.
I hope this hint is useful! (I can give you a hint for the hint if you'd like!)