Let $F, G : M \to N$ be diffeomorphisms of compact, connected, oriented, $n$-manifolds. If $F$ and $G$ are smoothly homotopic, prove that $F$ and $G$ are either both orientation-preserving or both orientation reversing.
Degree is homotopic invariant, hence $F$ and $G$ has the same degree, but how can I proceed?
To complete the proof with Henry's help in the answer as well as in the comment:
Lemma.
If $f: M \longrightarrow N$ is a diffeomorphism of smooth manifolds, then $\deg(f) = 1$ if $f$ is orientation preserving, and $\deg(f) = -1$ if $f$ is orientation reversing.Proof: Because $f$ is a diffeomorphism, for each $y$, there is only one $x$ such that $f(x) = y$. Hence, by
$$\deg(f) = \sum_{x \in f^{-1}(y)} \operatorname{sgn} (\det(df_x)),$$
where $y \in N$ is a regular value of $f$, we know that $\deg(f) = \pm 1.$
We know that degree is homotopic invariant, hence $F$ and $G$ has the same degree. Therefore, $F,G$ are both either orientation preserving or both orientation reversing.
Thanks very much, Henry!!!
Best Answer
To finish the problem, prove the following simple lemma:
Combining this with your observation about homotopy invariance of degrees, you see that $\deg(F) = \pm 1 = \deg(G)$, so they are both either orientation preserving or both orientation reversing.