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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.

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    $\begingroup$ If you take the cones over two distinct knots in $S^3$ you get two non-isotopic embedding of $D^2$ into $\mathbb R^4$. $\endgroup$
    – Bruno Martelli
    Commented May 12, 2021 at 14:45
  • $\begingroup$ Hmm. Yes. The issue is, why are they non-isotopic? I think what you are driving at is the following delicate question: given a PL-embedded disk $D$ (or surface) and a point $P$ on the disk, to what degree is the link of $P$ well-defined. More precisely, if you have two PL-balls $E$ and $F$ around $P$, are the isotopy types of $D\cap \partial E$ and $D\cap \partial F$ the same? $\endgroup$
    – Stefan Friedl
    Commented May 12, 2021 at 16:02
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    $\begingroup$ Usually the book of Rourke-Sanderson is a very useful reference for proper statements. It's only 120 pages, but it contains a lot of foundation stuff. I am grateful this book exists, especially now, since it seems to me that PL topology is not so widely well-known as it was before (although it is very powerful, in my opinion). $\endgroup$
    – Bruno Martelli
    Commented May 12, 2021 at 17:19
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    $\begingroup$ It turns out that it follows from the discussion on page 51 of Rourke-Sanderson that the cones on different knots are indeed inequivalent. So now I am completely satisfied. $\endgroup$
    – Stefan Friedl
    Commented May 14, 2021 at 16:23
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    $\begingroup$ Just a remark that Hudson is also often a very useful reference about the PL category. Of course Rourke-Sanderson is also great, but I've sometimes needed results they don't have that Hudson does, especially about noncompact things. $\endgroup$
    – Greg Friedman
    Commented May 15, 2021 at 4:21

2 Answers 2

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The precise result you want is Theorem 4.20 (page 56) in Rourke-Sanderson's book.

enter image description here

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.

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  • $\begingroup$ I'm not sure this quite addresses the original question, since in this theorem it seems that both I and M are n-dimensional but the question asks for the dimension of the domain ball to be smaller than the dimension of M. $\endgroup$
    – Greg Friedman
    Commented May 15, 2021 at 4:19
  • $\begingroup$ @GregFriedman: I think the point is that if $R$ is a regular neighborhood of an locally flat $k$-disk $D$, then $(R, D)$ is a standard ball pair. I agree that this requires a proof, so I was hasty. I will search for a better reference. $\endgroup$
    – Igor Belegradek
    Commented May 15, 2021 at 13:00
  • $\begingroup$ I see. But I don't think the original question requires the embedding to be locally flat. In fact, it looks like it's been determined in the comments that the lack of local flatness does pose an obstruction. $\endgroup$
    – Greg Friedman
    Commented May 17, 2021 at 2:55
  • $\begingroup$ @GregFriednab This just means that one has to require local flatness. $\endgroup$
    – Igor Belegradek
    Commented May 17, 2021 at 3:05
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Inspired by Bruno's comment I think it would be interesting if we consider the following more general result, which is Theorem 4.4.2 in [Daverman-Venema's book].

Theorem. Suppose $X$ is a compact, $k$-dimensional space and $M$ is a PL $n$-manifold, where $n \geq 5$ and $2k + 2 \leq n$. If $\lambda_0, \lambda_1 : X \to M$ are two topological embeddings whose images are 1-LCC in $M$ and $f_t : X \to M$ is a homotopy between $\lambda_0$ and $\lambda_1$, then there exists a compactly supported ambient isotopy $\Psi$ of $M$ such that $\Psi_0 = \text{Id}_M$ and $\Psi_1\lambda_0 = \lambda_1$.

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