Linear embeddings are simplicial?

Question

Let $K,L$ be finite simplicial complexes. Suppose there is a topological embedding $f: |K| \to |L|$ such that $f$ restricted to simplices of $K$ is linear (in particular $f(S)$ is completely inside a single simplex of $L$ for a simplex $S \in K$).

Is it always possible to find a subdivision of $L$, such that $f$ becomes simplicial? If not, what if $K$ and $L$ are PL-manifolds (with boundaries)? What is a good reference?

Why should it be true that $f(S)$ is contained in a single simplex? — Daniel Robert-Nicoud, Oct 03 '16 at 00:23
What are you using as your reference for the definitions of simplicial complexes and their geometrical realisations? — Rob Arthan, Oct 03 '16 at 01:07

score 0 · Accepted Answer · answered Aug 28 '19 at 12:40

Yes, it's always possible. Take a simplex $S$ of $K$ and subdivide the simplex of $L$ containing $f(S)$ so that $f(S)$ is a simplex of that subdivision (call the subdivision $L'$). For any other simplex $R$, $f(R)$ intersects the new simpleces of $L'$ linearly, and therefore $S'$ can be easily subdivided so that each simplex of the subdivision lies in a single simplex of $L'$ and we can continue by induction on $n$, where $n$ is the number of simpleces of $K$.

Linear embeddings are simplicial?

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