Just a short question about the degree of a homeomorphism. So, I understand that in the continuous setting we define the degree of a map $\ f: M \rightarrow M$ on a connected orientable manifold as the induced map on the top homology group (which is isomorphic to $\mathbb{Z}$).
If $\ f$ is a homeomorphism the degree can be either equal to $1$ or $-1$.
If the degree of a homeomorphism $f$ is equal to $1$ we call $\ f$ orientation-preserving.
Is it true that, since the degree respects the composition of maps, i.e. deg$(g \circ f) = $ deg($f$)deg($g$),
$f^2$ is always orientation-preserving for every homeomorphism $f:M \rightarrow M$?
Yes. You already gave the proof.