# algebraic topology – CW structure from map homotopic to the identity?

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?