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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.

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  • $\begingroup$ What is $F$? Should this be just $f$? $\endgroup$ – Benjamin Sep 11 '15 at 19:22
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    $\begingroup$ @Benjamin: Fixed the typo. Thanks! $\endgroup$ – Lisa Sep 11 '15 at 19:28
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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 $(x,t)$.

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