This is a follow-up concerning this question : Explaination concerning the polygonal generalized Schoenflies problem
I have trouble understanding the implication of the Shoenflies problem. I am currently working on ways to describe polygonal meshes, and I wanted to prove the floowing (intuitive) proposition : "Node positions and "cell-relationships" fully dertermine a polygonal embedding as long as cells of dimension $p$ have their nodes in an affine hyperspace of dimension $p$".
By "cell-relationship", I mean knwowing the relationship "is in the boundary of ..." acting on cells. The number of cells can be constrained to be finite if needed.
In my previous post, I formulated the questions in thechnical terms that I do not fully understand, and while it received a very good answer very quickly, I am still not sure about what it implies for my proposition.
So my question is the following : for dimensions $<4$, is the inductive proof using the Shoenflies theorem valid ? Can the theorem be saved for dimensions $\ge4$ ?
Thanks in advance,