$1_{S^{n-1}} \simeq$ to a constant map

Question

I have to show that $ \exists\ \ f : D^n \rightarrow S^{n-1} $ with $f\circ i =1_{S^{n-1}} \iff 1_{S^{n-1}} $ is homotopic to a constant map.

I don't know how to prove this. So, please help me in proving the above question.

Thanks!

Answer 1

Let $H: S^{n-1}\times [0,1]\to S^{n-1}$ be the homotopy such that $H(\cdot,1)$ is the identity and $H(\cdot,0)$ is constant. Then you define $f(r\, x):=H(r,x)$ for $r\in[0,1]$, $x\in S^{n-1}$. This is well-defined in $r=0$, as $H(\cdot ,0)$ is constant.

If $f$ is continuous and $f\circ i=1$, then you define $H(r,x):=f(r\,x)$.