TITLE: PL-embeddings of balls into PL-manifolds
QUESTION [8 upvotes]: Let $B$ be $k$-dimensional PL-ball and let $M$ be a connected  $n$-dimensional PL-manifold, let's say without boundary. Furthermore let $f,g\colon B\to M$ be two PL-embeddings.
If $k=n$, then the disc theorem (see Theorem 3.34 in Rourke-Sanderson) says that, after possibly composing $f$ by a reflection, $f$ and $g$ are ambiently isotopic.
Now assume that $k<n$. Is it true that $f$ and $g$ are necessarily ambiently isotopic?
If we work in the smooth category, then the answer would be yes. But I am very unsure about the PL-category.

REPLY [3 votes]: The precise result you want is Theorem 4.20 (page 56) in Rourke-Sanderson's book.

The notations in the statement are on page 50, but they are self-explanatory, e.g., $I^{n,q}$ is the standard disk pair ($n$-disk, $q$-disk), and you are assuming that $M^{n,q}$ is a closed manifold.