My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove?


It seems like it should be easy because the disk is compact and contractible, but I sill don't know what to do about it. I am working on problems in mathematical origami and I'm hoping to use this fact as a starting point in a much bigger proof.


This is not my area of expertise, so any tips on what to search or how to go about proving (or disproving) this would be helpful. I have searched different ways of wording this question and haven't found anything helpful.


I suspect this may generalize to closed $n$-balls embedded in $R^{n+1}$ or even an $(n+1)$-manifold, but I only need the case $n=2$.


Update: After researching some more based on ARupinski's comment, I realized that I am concerned specifically with piecewise-linear embeddings of the disk into $R^3$ with finitely many pieces. I don't think you can make a piecewise-linear Alexander Horned Shere with only finitely many pieces. So my question would be, if $f$ is a piecewise-linear embedding of the closed 2-ball $B^2$ into $R^3$ with finitely many pieces, and $g$ is another embedding of $B^2$ into $R^3$ such that $g[B^2]$ is a closed triangle (with interior), is there an isotopy between $f$ and $g$ using piecewise-linear embeddings along the way? This sounds a bit easier to go about proving... if it's true... :( This is somewhat related to things like the Carpenter's Rule Theorem for chain linkages in the plane.



Isotopy class of closed 2-ball embedded in R^3
The answer to your updated question is yes. In fact more is true:
any two PL embeddings $f,g$ of the 2-disk in any connected 3-manifold $M$ are isotopic.


One way to see this is to use the following statement, which follows from
Guggenheim's theorem: any two PL embedded $n$-balls in a connected PL
$n$-manifold are isotopic. (Guggenheim's theorem is slightly stronger in that it gives an *ambient* isotopy.)


The deduction is as follows: let $D\_1$ (resp. $D\_2$) be the image of
$f$ (resp. $g$). For $i\in \{1,2\}$ by thickening $D\_i$ you can construct an embedded 3-ball $B\_i$ containing $D\_i$ such that $\partial D\_i=\partial B\_i\cap D\_i$ and $D\_i$ is isotopic rel boundary to any of the two 2-disks contained in $\partial B\_i$ and bounded by $\partial D\_i$. By Guggenheim's theorem in dimension 3, we may assume that $B\_1=B\_2$. By the same theorem in dimension 2 we may assume that $\partial D\_1=\partial D\_2$, hence $D\_1$ is isotopic to $D\_2$.


You are asking for something stronger: namely $f,g$ should be isotopic,
not just their images. By the previous argument we can assume that $D\_1=D\_2$. Now use the fact that any orientation-preserving self-homeomorphism of the disk is isotopic to the identity (if the orientations are wrong, just flip one of the disks; this can be
achieved by an isotopy.)


A good book on elementary PL topology is *Introduction to piecewise linear Topology* by Rourke and Sanderson.


