A knot to me is the image of a smooth (pl locally flat) embedding $S^n \to S^m$ under the equivalence of smooth (ambient pl) isotopies. And more generally embeddings of closed manifolds $N \to M$.

These (pl vs smooth) objects are different in codimension $>2$: Haefliger knots are non-trivial while any pl-knot is trivial in this codimension. What about the case of $m-n=2$?

Triangulating smooth knots and isotopies seems to be doable, but can we smooth out pl knots and isotopies at least for $m=4, n=2$? Is there a reference?


1 Answer 1


PL locally flat knots have rank 2 $PL(2)$ normal bundles, but $PL(2)$ can be reduced to $SO(2)$ structure group. So there is a PL embedding of the smooth manifold $\nu S^2 \cong S^2 \times D^2 \hookrightarrow S^4$. This induces a smooth structure on a neighbourhood of the knot, namely the image of $\nu S^2$. Since the embedding is PL, there is a PL structure on the exterior of the knot $S^4 \setminus \nu S^2$, and the PL structures on the intersection $S^1 \times S^2$ coming from the exterior and coming from the smooth structure on $\nu S^2$ agree. Then use: given a PL structure on a 4-manifold with boundary $X$, together with a smooth structure on $\partial X$ refining the PL structure, there is (up to isotopy) a unique smooth structure on $X$ refining the PL structure (and restricting to the given smooth structure on $\partial X$, but this is automatic by uniqueness of smooth structures on 3-manifolds). This extends the smooth structure on $\nu S^2$ to all of $S^4$. Since this smooth structure on $S^4$ refines the standard PL structure on $S^4$, it is isotopic to the standard smooth structure on $S^4$. This isotopes the knot to a smooth embedding in the standard smooth structure.

Partial attempt at references. PL bundles: Rourke-Sanderson, Kuiper-Lashof. PL $\Rightarrow$ smooth for 4-manifolds: the key is Cerf's result $\Gamma_4=0$. Main reference is Hirsch-Mazur, Smoothings of Piecewise Linear Manifolds,

  • $\begingroup$ The normal structure is governed by the group PL_{4,2} of the germs of PL homeomorphisms of $R^4$ preserving $R^2\subset R^4$ pointwise. $\endgroup$
    – Victor
    Commented Jul 15, 2023 at 19:32

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