The general position theorem asserts that any $M$ $m$-manifold unknots in $R^n$ provided $n\geq 2m+2$. The general position theorem assumes a smooth setting. Is unknotting still hold in the PL setting? what is the lower bound on n in that case? what is the argument for unknotting a general $m$-manifold (compact, closed, not necessarily connected ) in the PL setting?



Unknotting a compact manifold in the PL setting
Yes, if two PL-embeddings $f,g:M^k\hookrightarrow N^n$ of a compact $PL$ manifold of dimension $k$ are homotopic and $n\ge 2k+2$, then they are PL-isotopic.


This is Corollary 5.9 in 


*Rourke, C. P.; Sanderson, B. J.*, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 69. Berlin-Heidelberg-New York: Springer-Verlag. VIII,123 p. with 58 fig. Cloth DM 42.00; $ 13.40 (1972). [ZBL0254.57010](https://zbmath.org/?q=an:0254.57010)


and a strengthened version is Theorem 4.1.1 in 


*Daverman, Robert J.; Venema, Gerard A.*, Embeddings in manifolds, Graduate Studies in Mathematics 106. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3697-2/hbk). xvii, 468 p. (2009). [ZBL1209.57002](https://zbmath.org/?q=an:1209.57002).


