Suppose $f:\Bbb R^n\rightarrow \Bbb R^n$ is a diffeomorphism. Let $D^n$ be the disk of radius $1$ centered at the origin.
My question is, is there an isotopy $g:\Bbb R^n\times I\rightarrow\Bbb R^n$ such that $g(\cdot,0)\upharpoonright D^n$ is the identity and $g(\cdot,1)\upharpoonright D^n$ is an homeomorphism onto $f(D^n)$.
If this does not hold, does it hold for the open ball centered at the origin instead?
Assume without loss of generality that $f(0) = 0$. Then $f$ is isotopic to a linear map via $F_t(x) = f(tx)/t$, and the result for $F_0$ follows from the fact that $\pi_0 GL_n(\mathbb{R}) = \{\pm 1\}$.