So I was looking at the proof given in Bott, Tu "Differential Forms in Algebraic Topology" of how to approximate continuous mapping by smooth mappings between manifolds. It is Proposition 17.8 on Page 213.
The Proposition goes as:
Every continuous map $f:M\rightarrow N$ between two manifolds is continuously homotopic to a differentiable mapping.
The proof basically goes as: First assume $N=\mathbb{R}$, then it is more or less obvious. Then for general $N$, by invoking Whitney Embedding Theorem, $N$ can be embedding into a $\mathbb{R}^k$ when $k$ is big enough. That is, $\exists g:N\rightarrow \mathbb{R}^k$smooth. Then the author made a remark which I don't see why:
If $g\circ f:M\rightarrow g(N)\subset \mathbb{R}^k$ is homotopic to a differentiable mapping, so is $f=g^{-1}\circ (g\circ f):M\rightarrow N$.
Here is the part I don't see why. How does the the author guarantee that the differentiable mapping homotopic to $g\circ f$ won't assume value in $\mathbb{R}^k\setminus g(N)$, thus making it impossible to take it back by applying $g^{-1}$.
I hope I am clear enough about my question. Any comment or reference to proofs of the same theorem would be appreciated!
Thanks!
Best Answer
Hopefully this is helpful. It is an excerpt from Lee's Introduction to Smooth Manifolds.