Let $M\_1$ and $M\_2$ be connected smooth manifolds and let $f\_0,f\_1:M\_1 \rightarrow M\_2$ be homotopic smooth maps such that some fixed point $p \in M\_2$ is a regular value for both $f\_0$ and $f\_1$. Question: Can I always find a smooth homotopy $F:M\_1 \times I \rightarrow M\_2$ such that $p$ is a regular value for $F$?


This question came up in the smooth manifolds course I'm teaching right now. If it had a positive (and relatively elementary) answer, then it would allow me to greatly simplify some of the proofs I'm getting ready to present.



Homotopies with prescribed regular values
The answer is "yes" if $M\_1$ is compact. Here's a sketch of a proof.


Consider any smooth homotopy $G:M\_1 \times I \rightarrow M\_2$ between $f\_0$ and $f\_1$. Since $M\_1$ is compact, the set of regular values of $f\_0$ and $f\_1$ are both open. Moreover, the condition of being a regular value is stable. We can thus find some $\epsilon>0$ and some tiny open ball $U$ around $p \in M\_2$ such that every point of $U$ is a regular value of the function $g\_t:M\_1 \rightarrow M\_2$ defined via $g\_t(x) = G(x,t)$ for all $t \in [0,\epsilon] \cup [1-\epsilon,1]$.


Using Sard's theorem, we can find some $q \in U$ such that $q$ is a regular value of $G$. Choose some smooth function $H:M\_2 \times I \rightarrow M\_2$ with the following properties:


1. For $0 \leq t \leq 1$, the function $h\_t:M\_2 \rightarrow M\_2$ defined via $h\_t(x) = H(x,t)$ is a diffeomorphism supported on $U$.
2. $h\_0 = \text{id}$ and $h\_1(q)=p$.
3. $h\_{t} = h\_{t'}$ for all $t,t' \in I$ that are sufficiently close to $1$.


Now define a smooth function $F:M\_1 \times I \rightarrow M\_2$ via the following formula:


$$F(x,t) = \begin{cases}
G(h\_{t/\epsilon}(x),t) & \text{if $t \in [0,\epsilon]$},\\
G(h\_1(x),t) & \text{if $t \in (\epsilon,1-\epsilon)$},\\
G(h\_{1-(t-1+\epsilon)/\epsilon}(x),t) & \text{if $t \in [1-\epsilon,1]$}.
\end{cases}$$


Thus $F$ is a smooth homotopy from $f\_0$ to $f\_1$. The third condition above is needed to ensure that $F$ is smooth. I claim that $p$ is a regular value of $F$. Indeed, consider $(x,t) \in F^{-1}(p)$. There are three cases:


1. If $t \in (\epsilon,1-\epsilon)$, then $(x,t) \in G^{-1}(q)$ and $F$ agrees with $G \circ h\_1$ in a neighborhood of $(x,t)$. Since $q$ is a regular value of $G$, it follows that $G$ is a submersion at $(x,t)$, and thus since $h\_1$ is a diffeomorphism it follows that $F$ is also a submersion at $(x,t)$.
2. If $t \in [0,\epsilon]$, then setting $f\_t(x) = F(x,t)$ we have that $f\_t = g\_t \circ h\_{t/\epsilon}$. Since $h\_{t/\epsilon}$ is a diffeomorphism and $h\_{t/\epsilon}^{-1}(p) \in U$ and every point of $U$ is a regular value for $g\_t$, it follows that $f\_t$ is a submersion at $x$. This implies that $F$ is a submersion at $(x,t)$.
3. If $t \in [1-\epsilon,1]$, then an argument similar to the previous one shows that $F$ is a submersion at $(x,t)$.


