Equivalence of homotopy, relative homotopy and continuous extension

Question

Let $ X $ be a topological space, $ a\in S^1 $ and $ f:\ S^1\rightarrow X $ a continuous map. Show that the following statements are equivalent:

(a) $ f $ and a constant map $ g $ are homotopic relative to $ \{a\} $.

(b) $ f $ is homotopic to a constant map $ g $.

(c) $ f $ can be continuously extended to a continuous map $ \tilde{f}:\ D^2\rightarrow X $.

(a) to (b) is trivial. However, I don't know how to show (b) to (c) and (c) to (a). Unfortunately, I can't even show any progress so far. I have no idea how to approach this proof. Maybe someone can help.

Thanks a lot in advance!

Answer 1

$b) \Rightarrow c)$

Let $f$ be homotopic to a constant map, i.e. there is a homotopy $H:S^1\times[0,1]\to X$ such that $H(\cdot, 0) = f$, $H(\cdot,1) = g$ where $g = \operatorname{const}$.

Since $H$ is constant for $t = 1$, it induces a continous map $D^2 \simeq (S^1\times[0,1]/S\times\{1\}) \to X$.

Since $f:S^1 \to X$ is defined on the boundary $\partial D^2$, $\overline{f}:D^2 \to X$ is a continous extension of $f$, that is $$\overline{f}|_{S^1} = f$$ thus $f = \overline{f} \circ i$ where $i:S^1 \hookrightarrow D^2$ denotes the inclusion map.