PL embeddings and subpolyhedra

Let $P$ and $Q$ be compact piecewise-linear polyhedra. A piecewise-linear map $f:P\to Q$ is a map that is simplicial after suitable finite subdivisions of triangulations of $P$ and $Q$. A subpolyhedron $A$ of $Q$ is a subset represented by the realization of a subcomplex in some finite linear triangulation of $Q$. A map $f:P\to Q$ is a piecewise-linear embedding if it is piecewise-linear, its image $f(P)$ is a subpolyhedron of $Q$, and $f:P\to f(P)$ is a piecewise-linear homeomorphism. A finite simplicial triangulation of a closed piecewise-linear $n$-manifold is combinatorial if the link of every $k$-simplex is a piecewise-linear sphere of dimension $n-k-1$. Thus in a closed combinatorial $3$-manifold every triangular face has link $S^0$ and is incident to exactly two tetrahedra.

Subcomplexes, closures, stars, and links in a simplicial complex — a subcomplex is closed under taking faces.

Rourke–Sanderson, §§1.4, 1.7, 2.14 — piecewise-linear maps, embeddings and their simplicial subdivisions.