I am a little lost with the definitions of simplicial subcomplexes and linear embeddings, and how they compare to each other. More precisely, I do not understand why the following theorems are not contradictory.
Brehm and Sarkaria showed that for every $d\geq 2$, and $k$ with $d+1\leq k\leq 2d$, there is a finite $d$-dimensional simplicial complex $X$ which admits a piecewise linear embedding in $\mathbb{R}^k$ but not a geometric/linear embedding, see also this answer.
Adiprasito and Patakov showed the following "Kind-of Fary's theorem":
Consider $X$ a finite simplicial complex and a PL embedding $\phi:X\to M$, where $M$ is a PL-manifold. Then there is a triangulation $M'$ of $M$ that contains $X$ as a subcomplex.
Since $M := \mathbb{R}^k$ is a PL-manifold, this theorem implies that any PL embedding of $X$ into $\mathbb{R}^k$ yields a triangulation of $\mathbb{R}^k$ containing $X$ as a subcomplex. Together with the first theorem, this means that there exist simplicial complexes that are subcomplexes of a triangulation of $\mathbb{R}^k$ but do not embed linearly into $\mathbb{R}^k$.
My question is: What does it mean that a triangulation of $\mathbb{R}^k$ contains $X$ as a subcomplex? Or, how can it be that $X$ being a subcomplex does not give rise to a linear embedding of $X$ in $\mathbb{R}^k$?
