Homeomorphisms between two punctured planes

Question

Question:

Let $p,q$ be two different points in the interior of $D\subset\Bbb{R}^2$ where $D$ is the closed unit disk. Is there a homeomorphism $h:\Bbb{R}^2\setminus\{p\}\to\Bbb{R}^2\setminus\{q\}$ such that $h|_{\partial D}=id_{\partial D}$?

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This question is motivated by the following exercise:

Let $D\subset\Bbb{R}^2$ be the closed unite disk $\{x\in\Bbb{R}^2\mid \|x\|=1\} $ and $f:D\to\Bbb{R}^2$ a continuous map such that $f|_{\partial D}=id_{\partial D}$. Show that $D\subset f(D)$.

which looks very similar to the following ones:

Continuous function from the closed unit disk to itself being identity on the boundary must be surjective?

Let $f : D \rightarrow D$ be a continuous map whose restriction to $S^1$ is the identity map. Show that $f$ must be surjective.

I tried to adapt the techniques in the answers to those two questions and came up with the question above that I don't really see how to go on.

Answer 1

Yes. Consider a cone $C$ that intersects the plane in $\partial D$ and with vertex $t$. Then the projection along the line $tp$ gives us a homeomorphism $C\to\Bbb R^2$ with $t\mapsto p$; the same can be done with $t\mapsto q$. Combine these (and remove the points $p,t,q$):

Answer 2

Yes, take a non-self intersecting path $\gamma:[0,1]\rightarrow \mathbb{R}$ from $p$ to $q$ such that $\dot \gamma(t)\not=0$. Extend the $\dot \gamma$ to a vector field in a small neighborhood of the curve. Using a bumpfunction extend this vector field to a compactly supported vector field in the interior of $D$. The flow of this vector field at time $1$ will give you the required homeomorphism (diffeo even), as $\gamma$ is a solution curve. Note that you can extend this idea to many points, and also on other higher dimensional manifolds.