Every map $S^1 \rightarrow X$ is homotopic to a constant map $\rightarrow$ every map $S^1 \rightarrow X$ extends to a map $D^2 \rightarrow X$

Question

Every map $S^1 \rightarrow X$ is homotopic to a constant map $\rightarrow$ every map $S^1 \rightarrow X$ extends to a map $D^2 \rightarrow X$

I know this has done, but I wanted someone to verify my proof:

$proof$:

Let $f: S^1 \rightarrow X$ be a map

Let $F: S^1 \times I \rightarrow X$ be a homotopy from $f$ to the constant map, $c$ such that $F(x,1)=f(x)$ and $F(x,0)=c_0$.

Define $\gamma: S^1 \times I \rightarrow D^2$ by:

$\gamma(x,s) \rightarrow s(cos(x)+sin(x))$

Then $f$ is constant on the fibers of $\gamma$ and so induces a map $\tilde{f}: D^2 \rightarrow X$ that extends $f$.

Is this correct?

Answer 1

You need instead to define $\gamma:D^2\rightarrow S^1\times I$ by using polar coordinates: $\gamma(r,\theta)=(e^{i\theta},r)$, and you can define $F\circ\gamma$.