# Unknotting a compact manifold in the PL setting

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?

## 1 Answer

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; $13.40 (1972). ZBL0254.57010 and a strengthened version is Theorem 4.1.1 in Daverman, Robert J.; Venema, Gerard A., Embeddings in manifolds, Graduate Studies in Mathematics 106. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3697-2/hbk). xvii, 468 p. (2009). ZBL1209.57002. • Can corollary 5.9 be generalized to: if two PL-embeddings$f,g : M_1^k1 \sqcup M_2^{k2} \longrightarrow N^n$are homotopic and$n\geq k_1+ k_2+2$then they are PL-isotopic ? (here$M_i^{k_i}$is a compact PL$k_i$manifold ) – Steve May 24 '19 at 14:09 • @Steve: I would say no. Note that the dimension of the disjoint union is$\operatorname{max}\{k_1,k_2\}$, and in general$k_1+k_2<2\operatorname{max}\{k_1,k_2\}$. In particular if$M^1=S^1$and$M_2=\varnothing$(regarded as a$0$-manifold, say) then with$N=S^3$you get classical knot theory. – Mark Grant May 24 '19 at 14:39 • thank you. So the result holds with$n\geq max\{k_1,k_2\}+2\$. Just to clarify, by PL-isotopic, do you mean PL-ambient isotopic? or something else ? – Steve May 24 '19 at 18:31
• @Steve: I think you need a factor of 2 in front of the max. And I mean whatever is stated in those textbooks, I think it's ambient PL-isotopy. – Mark Grant May 24 '19 at 19:12
• Sure, dropping the 2 was a typo. thanks again! – Steve May 24 '19 at 19:14