Let $M,N$ be compact connected manifolds, $f:M \to N$ a smooth map with $\operatorname{rank}{(df)}=\dim{N}$. Then for all points $p,q \in N$ ; $f^{-1}p$ is diffeomorphic to $f^{-1}q$.
Please help me solve this question, I've no idea.
Let $M,N$ be compact connected manifolds, $f:M \to N$ a smooth map with $\operatorname{rank}{(df)}=\dim{N}$. Then for all points $p,q \in N$ ; $f^{-1}p$ is diffeomorphic to $f^{-1}q$.
Please help me solve this question, I've no idea.
Fix $p \in N$. Since $N$ is connected, it suffices to show that the set $S := \{q \in N : f^{-1}(p)\simeq f^{-1}(q) \}$ is opened and closed.
From the response of a question on MSE, we know that $S$ is opened. If $q \in \bar{S}$, let $U \subset N$ be a neighborhood of $q$ on which all fibers are diffeomorphic to $f^{-1}(q)$. Since $U$ contains some point in $S$, we conclude that $q \in S$, hence $S$ is closed.