I am basic with homotopy theory and especially with CW-approximation, Postnikov and Whitehead towers. In the proof of such things one need the following result on and on:
Let $X'$ be obtained from $X$ by attaching a $(n+1)$-cell, i.e. we have a pushout diagram of the following from:
$S^n\ \ \stackrel{f}{\rightarrow}\ \ X$
$\downarrow\ \ \ \ \ \ \ \ \ \ \ \ \downarrow\ i$
$D^{n+1}\rightarrow X'$
Let $\alpha=[f]$. Then there holds the following for the induced map $i_*$: $i_*(\alpha)=0$. In particular, $\alpha$ is killed.
I don't see why this should be true. Is there someone who can help me with the proof of this result? Thanks.
Start with the fact that there is a homotopy $h : S^n \times [0,1] \to D^{n+1}$ from the inclusion map $S^n \hookrightarrow D^{n+1}$ to a constant map. Next, letting $g : D^{n+1} \to X'$ be the characteristic map on the bottom arrow of your diagram, the composition $g \circ h : S^n \times [0,1] \to X'$ is a homotopy from $i \circ f$ to a constant map.