Suppose $X$ is a $CW$ complex, and $h:Xto X$ is a (cellular) map which is homotopic to the identity.

What can one say about the the image $h(X)$? In particular I am hoping to give it some natural CW structure, hopefully homotopic to X.

As an example: if $A$ is a contractible subcomplex, applying the homotopy extension property gives a map $h:Xto X$ with $h(A)= point$, and the image of $h$ is ‘morally’ $X/A$ and homotopic to $X$, although this is a bit tricky to prove, and involves $A$ being contractible.

If $X$ is $k$-connected, the inclusion of the $k$-skeleton is nullhomotopic, we can apply the homotopy extension principle to get $h:Xto X$, and I would really like to conclude $X/X^{k}$ is homotopic to $X$. This is not true as written, but $X$ is homotopic to a complex with trivial $k$-skeleton. I know this can be proved with Whitehead’s theorem, but can someone help me get a more ”barebones” picture along the lines above?