In my quest to compute the fundamental group of some object in multiple ways, questions concerning cellular approximations of homeomorphisms came up. The setting is as follows:
Suppose $M$ is a connected, compact (but possibly with bounday) surface (2-manifold). By the triangulation theorem for surfaces, $M$ is homeomorphic to some polyhedron, which ought to induce a natural CW complex structure on $M$, if I am not mistaken. Thus, we can consider a given homeomorphism, which we also assume to be orientation-preserving, $f:M\to M$ to be a self-homeomorphism of a CW complex. By the cellular approximation theorem, $f$ is homotopic to some cellular map $g:M \to M$.
Now I have some related questions concerning cellular approximations of $f$:
1.) Can we make sure that we homotope $f$ to a cellular map $g$ that is also a homeomorphism?
2.) Suppose that we can indeed homotope to a cellular homeomorphism $g$. Collapse any 1-cell in $M$, and call the resulting complex $N$, the canonical projection being $\pi:M \to N$. If we set $g'(x) = \pi \circ g (z)$ where $z$ is any element in the fiber of $x$ under $\pi$, will $g'$ be a: a) a well-defined map? b) continuous? c) even a homeomorphism?
I would already be extremely glad about any simple "yes or no"-answers, as I would just like to know whether this track is worth pursuing at all. I have a feeling that these questions should have affirmative answers (just thinking of simplical maps, for instance), but I am always cautious about gut feelings.
This is NOT a homework question (not for or any person I know, at least), but rather something that came up in my independent studies.
Thank you in advance!