Smooth homotopy

Question

Let $M,N$ be manifolds. Suppose that $f_0, f_1:M\stackrel{C^\infty}\to N$ are homotopic, i.e. there exists a continuous mapping $f:M\times[0,1]\to N$ s.t. $f(x,0)=f_0(x)$, $f(x,1)=f_1(x)$.

Then is there any smooth homotopy $F\in C^\infty(M\times[0,1],N)$ connecting $f_0$ to $f_1$? How to construct it?

Answer 1

Yes. I assume you mean $M$ and $N$ to be manifolds without boundary, though I expect the result is true when they are, too.

This follows from the fact that, given a manifold with (or without) boundary $f: (W, \partial W)$, a manifold without boundary $N$ and a map $(W, \partial W) \to N$ that's smooth on the boundary, $f$ is homotopic to a smooth map by a homotopy that doesn't modify the map on the boundary. This is known as the Whitney approximation theorem; you can find a proof in e.g. Lee's "Introduction to Smooth Manifolds". Simply take $W = M \times I$ and $f = f_t$; then by hypothesis $f|_{\partial W}$ is smooth.

Note that if we take $W = M$, we get that every continuous map is homotopic to a smooth one; so $[M,N]$ and $[M,N]_{\text{smooth}}$ are in bijection. So there's no interesting difference between "continuous" and "smooth" homotopy theory.

Answer 2

The Whitney Approximation Theorem says that for smooth manifolds $X$ and $Y$ (with $\partial Y = \emptyset$), any continuous map $X \to Y$ is (continuously) homotopic to a smooth map $X \to Y$. Moreover, we can choose the homotopy to leave the map alone wherever it is already smooth. So in your case there is a smooth homotopy $F: M \times [0,1]\to N$ that is (continuously) homotopic to $f$ and satisfies $F_0=f_0$ and $F_1=f_1$.

Answer 3

This problem seems to have already been solved but, ...

I found clear statement in the authoritative literature, I will introduce it for your reference.

The first part of the following proposition seems to be the answer to your problem. See the P258 of ref 1　( The second edition seems to be published now. I don't know which page has this proposition in the second edition.).

As your comments, we can consider “both ∂M and ∂N are an empty set”. Therefore, the first half of the following proposition should be the perfect answer.

Reference:
(ref.1)John M. Lee; "Introduction to Smooth Manifolds (Graduate Texts in Mathematics, 218)" Springer (2002/9/23)

P.S:
The first half of this proposition seems to give a complete answer to your question.　However, it seems that we have to do a little work in case of my problem.

I'm not very good at English, so I'm sorry if I have some impolite or unclear expressions. I welcome any corrections and English review. (You can edit my question and description to improve them)