Suppose $X$ is a $CW$ complex, and $h:X\to X$ 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 $h:X\to X$ 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 $h:X\to X$, 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?
Best Answer
It does not work in general. Let $X = D^2$ = plane disk. It contains the Hawaian earring $H$ which is no CW complex (it does not even have the homotopy type of a CW complex). We may assume that the biggest circle in $H$ is $\partial D^2$.
Give $X$ the CW structure with one $0$-cell (consisting of a point $* \in \partial D^2$), one $1$-cell and one $2$-cell. Then $X^1 = \partial D^2$.
Let $f : D^2 \to [0,1]$ be a continuous surjection such that $f(X^0) = \{0\}$ and $f(X^1) = [0,1/2]$. We can easily construct a continuous surjection $g : [0,1] \to H$ such that $g(0) = *$ and $g([0,1/2]) = \partial D^2$. The map $h = g f : X \to X$ is cellular and homotopic to the identity. Its image is $H$.