Let $f:S^1 \to X$ a continuous function $X$ a topological space. Then $f$ is homotopic to a constant iff $f$ extends to $D$.

Question

Let $X$ a topological space, $D$ a open unitary disc on $\mathbb{R}^2$ and $S^1 = \partial D.$

How to show that $f: S^1 \to X$ continuous is homotopic to a constant map iff there is a continuous extension $\bar{f} : D \to X$, I mean, $\bar{f}$ is continuous and $\bar{f}|S^1 = f.$

Answer 1

define $H(t,x):S^1\times I \rightarrow X$ $H(t,x)=\bar f(tx)$.

suppose $f$ is homotopic to a constant consider $H:S^1\times I\rightarrow X$, $H(0,x)=c$, $H(1,x)=f$, write $$\bar f:D\rightarrow X$$ $$\bar f(x)= H(\|x\|,{x\over {\|x\|}})~~\text{if}~ x\neq 0,$$ $$\bar f(0)=c.$$