I am trying to work through Hatcher's proof (page 344 of his book) that the homotopy sequence of a triple $$... \rightarrow \pi_n(A,B,x_0) \stackrel{i_*}{\rightarrow} \pi_n(X,B,x_0) \stackrel{j_*}{\rightarrow} \pi_n(X,A,x_0) \stackrel{\partial}{\rightarrow} \pi_{n-1}(A,B,x_0) \rightarrow ..$$ is exact. Specifically, I want to show that $\mathrm{Ker}(\partial) \subseteq \mathrm{Im}(j_*).$

If $f$ represents a class $[f] \in \mathrm{Ker}(\partial)$, then we choose a homotopy $F$ between $f|_{I^{n-1}}$ and a map with image in $B$. Hatcher says "we can tack $F$ onto $f$ to get a new map ... which... is homotopic to $f$ by the homotopy that tacks on increasingly longer initial segments of $F$". I don't understand this, and I haven't been able to write down any map $g$ with $j_*[g] = [f].$


As you can see in the picture below, I tacked the homotopy, which I named $H$, to $f$ which is possible since $H_0 = f$. Also, $\partial D^n$ is now completely mapped to $x_0$, which makes the constructed map $g$ one of the form $(D^n, \partial D^n, s_0) \to (X,x_0,x_0)$, as desired.

You desired map <span class=$g$">

To get $j_*[g] = [f]$, you push in/out the cylinder via a homotopy $G$ as in the remaining three pictures below.

G_0 G_{\frac{1}{2}} G_1


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