Let $X$ be a CW complex, and let $F:X\to X$ be an homeomorphism that sends each cell onto some cell; notice that we could say that $F$ is an automorphism of the CW complex since it preserves its cell decomposition.
Assume that $F$ fixes some cell $e$, that is, $F(e)=e$. Does $F$ necessarily have a fixed point?
So far, I've considered the following. Let $n$ be the dimension of $e$. If $n=0$, then the answer to the question is trivially yes. If $n=1$, the answer is also yes, because either the closure $\overline{e}$ of $e$ is homeomorphic to $[0,1]$, in which case we can use the Brouwer fixed theorem in a rather elementary way, or to a circle, in which case the only $0$-cell attached to the circle is necessarily fixed.
For $n>1$, the situation is not as obvious to me. If one assumes that the CW is contractible, there is a solution here, which apparently is related to the Lefschetz fixed point theorem, a generalization of the Brouwer fixed point theorem:
A version of Brower's fixed point theorem for contractible sets?
I'm not exactly sure how to tackle the general case. Any help is appreciated!
