Let $X$ be a CW-complex with skeleta $X_0 \subset X_1 \subset \cdots $
Let $\gamma \colon [0,1] \rightarrow X$ be a path between points $x,y \in X_0$.
Because $[0,1]$ is compact, Im$\gamma \subset X_n$ for some $n > 0$. (I think.)
I want to replace $\gamma$ with a path in $X_{n-1}$. I would like to maybe take $\gamma' = \gamma \cap X_{n-1}$ and somehow "complete" it.
If I then continue this process, I will end up with a path in $X_1$.
How can I do this?
Thanks very much for your help.
$\gamma$ is in fact homotopic, rel endpoints, to a path in the 1-skeleton. As you say, it certainly lies in the $n$-skeleton for some $n$. By a general position argument, if $n > 1$, you can alter $\gamma$ by a small perturbation so that it misses the center of each $n$-cell. Let's let $X^n$ denote $X_n$ without the centers of the $n$-cells. Then $X^n$ deformation retracts onto $X_{n-1}$ in general, just as a punctured disk retracts onto its boundary. By composing $\gamma$ with such a deformation, we homotop it into a path in $X_{n-1}$. Do this repeatedly until you get to $n = 1$, at which point the general-position argument fails, but that's OK, because you've got what you wanted.