Homotopy on $S^{n-1}$ extends to homotopy on $D^n$

Question

I have the following problem:

I have $\beta:D^n\to Z$ such that $\beta\circ i\simeq h$, where $i:S^{n-1}\hookrightarrow D^n$ is the inclusion and $h:S^{n-1}\to Z$ some map. Then is there a $\beta':D^n\to Z$ with $\beta\simeq\beta'$ such that $\beta'\circ i=h$?

I've looked for the answer a bit online, and I've understood that this condition is being a cofibration, but I didn't explicitly see the proof that $i:S^{n-1}\hookrightarrow D^n$ is a cofibration.

Answer 1

It clearly suffices to show that there exists a retraction $r : D^n \times I \to D^n \times \{0\} \cup S^{n-1} \times I$. In fact, the map $\beta$ and the homotopy $\varphi: \beta \circ i \simeq h$ can be pasted to a continuous $\phi : D^n \times \{0\} \cup S^{n-1} \times I \to Z$ and $\phi \circ r$ is what you desire.

Define $$r(x,t) = \begin{cases} (\frac{2x}{2-t},0) & \lVert x \rVert \le \frac{2-t}{2} \\ (\frac{x}{\lVert x \rVert},2 - \frac{2-t}{\lVert x \rVert}) & \lVert x \rVert \ge \frac{2-t}{2} \end{cases} $$ This is a well-defined continuous map because

1. $\lVert \frac{2x}{2-t} \rVert \le 1$ for $\lVert x \rVert \le \frac{2-t}{2}$

2. $1 \ge t \ge 2 - \frac{2-t}{\lVert x \rVert} \ge 0$ for $1 \ge \lVert x \rVert \ge \frac{2-t}{2}$

3. For $\lVert x \rVert = \frac{2-t}{2}$ both parts yield the same value.

For $t=0$ we get $\frac{2-t}{2} = 1$ and thus $r(x,0) = (x,0)$ for all $x \in D^n$. For $x \in S^n$ we have $r(x,t) = (x,t)$. Thus $r$ is a retraction.

There is a nice geometric interpretation of $r$: For each $(x,t)$ consider the line $L_{x,t}$ through $(0,2)$ and $(x,t)$. This line intersects $D^n \times \{0\} \cup S^{n-1} \times I$ in a single point $r(x,t)$.

Remark:

It is very easy to see that for a closed subspace $A \subset X$ the inclusion $i : A \to X$ is a cofibration if and only if $X \times \{0\} \cup A \times I$ is a retract of $X \times I$. See for example Retraction induces Cofibration . This is also true for non-closed $A \subset X$, but the proof is sophisticated. See

Strøm, Arne. "Note on cofibrations II." Mathematica Scandinavica 22.1 (1969): 130-142.