I am refering to this document about the Spanier-Whitehead Duality. The setting is that $X$ is a compact simplicial complex and $f,g\colon X \to S^n$ are two simplicial embeddings. In the first sentence of the proof, the author says:
If $n$ is sufficiently large relative to the dimension of $X$, then $f$ and $g$ are isotopic, so $S^n \backslash f(X)$ is homeomorphic to $S^n \backslash g(X)$.
I have the following questions:
- From what I understand, a simplicial embedding is an injective function between simplicial complexes such that there is an isomorphism between the $0$-simplices, simplices go to simplices, and a face of a simplex goes to a face of the same simplex. Is this definition correct? And in this case, is any embedding $f\colon X \to S^n$ a simplicial embedding? It does seem so to me, since I think we can create $S^n$ with as many simplices as we like.
- I can intuitively understand why $f$ and $g$ would be isotopic—there is enough space to 'move' $X$ around in $S^n$ if $n$ is large. Is there some theorem that would make this intuition more concrete?
- I do not understand why $S^n \backslash f(X)$ is homeomorphic to $S^n \backslash g(X)$.