# Can surfaces be interestingly knotted in five-dimensional space?

It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not.

Everybody loves knots — one-dimensional compact manifolds mapped generically into three-dimensional compact manifolds — and it's natural to ask about "knots" in higher dimensions. Of course, the space of generic maps of a one-dimensional compact manifold into a four-dimensional compact manifold is connected, so there is no interesting "knotting". Instead, people usually think about "surface knots in 4d", which are usually defined as embedded compact 2-manifolds in a 4-manifold.

But surfaces can map into 4-space in much more interesting ways. In particular, whereas a generic map from a 1-manifold to a 3-manifold is an embedding, two generic surfaces in 4-d can be "stuck" on each other: the generic behavior is to have point intersections. So a richer theory than that of embedded surfaces in 4-space is one that allows for these point self-intersections — it would be the theory of connected components of the space of generic maps.

Still, though, thinking about these self-intersections is hard, and their existence is part of what makes 2-knot theory hard (for instance, it interferes with developing a good "Vassiliev" theory for 2-knots). If you really want to reproduce the fact that generic maps have no self intersections, you should move the ambient space one dimension higher.

Hence my question:

Can compact 2-manifold embedded into a compact 5-manifold be interestingly "knotted"? I.e. let $L$ be a compact 2-manifold and $M$ a compact 5-manifold; are there multiple connected components in the space of embeddings $L \hookrightarrow M$?

I expect the answer is "no", else I would have heard about it. But my intuition is sufficiently poor that I thought it best to ask.

• Jan 20 '11 at 22:11
• and eom.springer.de/m/m065130.htm (I'm on an annoying portable device: sorry I can't be more verbose) Jan 20 '11 at 22:19
• @Mariano: Whereas, see, I googled for all sorts of things like "knots in five space" and "surface knots in five space" and read Wikipedia's entires that google turned up, which somehow missed that one. Jan 21 '11 at 4:09

If $M$ is a connected compact $2$-manifold, then it unknots in $\Bbb R^5$. More generally, $k$-connected $n$-manifolds embed in $\Bbb R^{2n-k}$, provided $k<\frac{n-2}2$, and unknot in $\Bbb R^{2n-k+1}$, provided $k<\frac{n-1}2$. This was proved around 1961 - by Roger Penrose, J.H.C. Whitehead, and Zeeman in the PL category; and by Haefliger in the smooth category. Later Zeeman and Irwin relaxed the metastable dimension restrictions in the PL result to codimension $\ge 3$ (see Zeeman's "Seminar on Combinatorial Topology").

On the other hand, the disjoint union of two $2$-spheres is a compact $2$-manifold. It definitely knots in $\Bbb R^5$ as detected by the linking number. That is the degree $\alpha$ of $S^2\times S^2\to S^4$, $(p,q)\mapsto \frac{f(p)-g(q)}{||f(p)-g(q)||}$, calling our link $f\sqcup g:S^2\sqcup S^2\to\Bbb R^5$. A nontrivial link is the Hopf link, whose components are the factors of the join $S^5=S^2*S^2$. Since $S^2$ unknots in $\Bbb R^5$, the exterior of one component is always homotopy equivalent to $S^2$, and the linking number is also the degree $\lambda$ of $p(S^2)\to S^5\setminus q(S^2)\simeq S^2$. [In different dimensions, where $\alpha$ and $\lambda$ are not numbers but homotopy classes of spheroids (more precisely $\alpha$ factors though a spheroid up to homotopy, upon killing the wedge), their relation is more interesting: $\alpha$ equals the suspension of $\lambda$ (up to a sign).]

By Haefliger's theorem (1963) that embeddings in the metastable range are classified by equivariant homotopy of two-point configuration spaces, the linking number for each pair of components is the only invariant of smoothly embedded $2$-manifolds in $\Bbb R^5$. This recovers the result that connected surfaces unknot in $\Bbb R^5$; and additionally implies that there is nothing new for $3$-component links. [In contrast, there are the Borromean rings of three $3$-spheres in $\Bbb R^6$, whose nontriviality is detected e.g. by a nonvanishing triple Massey product in the complement. Thinking of the usual Borromean rings in $\Bbb R^3$ as lying in the three coordinate planes, one can similarly do three copies of $S^1*S^1$ lying in the two-factor subproducts of $\Bbb R^2\times\Bbb R^2\times\Bbb R^2$.]

Also smooth, PL and topological knot theories coincide for smooth $n$-manifolds in smooth $m$-manifolds in the metastable range $m>\frac{3(n+1)}2$ (this includes $2$-manifolds in $\Bbb R^5$). In more detail, Haefliger's classification theorem implies that if two smooth embeddings in the metastable range are isotopic (=homotopic through topological embeddings, possibly wild) then they are smoothly isotopic. Weber's PL classification theorem (1967) implies additionally that every PL embedding of a smooth manifold in the metastable range is ambient isotopic to a smooth embedding. Also it follows from results of Edwards and Bryant that an arbitrary topological embedding in codimension $\ge 3$ is isotopic to a PL embedding, and, from results of Bryant-Seebeck, that a locally flat topological embedding in codimension $\ge 3$ is ambient isotopic to a PL embedding.

I will answer for embeddings of Sn in Sn+3.

No in the PL category by Zeeman.
No in the topological category by Stallings.
Yes in the differentiable category. See Levine and Haefliger.

EDIT: In the case $n=2$, which the question is about, the result is no in the differentiable category as well, as pointed out in other answers.

• The result by Stallings is about locally nice topological embeddings? Jan 20 '11 at 22:25
• @Mariano- Of course. Otherwise there would be knotting in any codimension due to the existence of wild embeddings. Jan 20 '11 at 22:32

An important reference concerning higher-dimensional knots is this paper of Zeeman of 1962. The paper considers the existence of knotted $k$-spheres inside $n$-spheres for general $n>k$. He proves that in the piecewise-linear setting a $k$-sphere in a $n$-sphere is always unknotted provided the codimension $n-k$ is strictly greater than 2, hence in particular there are no knotted 2-spheres in $S^5$ or $\mathbb R^5$ in the PL category.

As he says in the introduction of the paper, knotted spheres may exist however in the smooth category. The lowest-dimensional ones are some knotted $S^3$ inside $S^6$ found by Haefliger. However, Haefliger also shows that $S^k$ is always unknotted in $S^n$ when $n>3(k+1)/2$, and for $k=2$ we get $n>4.5$: hence 2-spheres unknot in $S^5$ or $\mathbb R^5$ also in the smooth category.

I don't know the proofs, but maybe one can prove that $S^2$ unknots in $S^5$ by decomposing $S^2$ as a 0- and 2-handle, thicken and extend this decomposition to the whole of $S^5$, and then simplify handles as in Smale's proof of Poincaré conjecture. By trading 1- and 4-handles with 3- and 2-handles we remain with 2- and 3-handles only, the core of each being attached to a (necessarily unknotted) circle in $S^4$ (the boundary of the 0-handle). The resulting decomposition of $S^5$ should be standard and hence the 2-sphere is unknotted (I guess).

I think the first reference in the literature for the result you want is Wen-Tsun Wu, On the isotopy of $C^r$-manifolds of dimension $n$ in Euclidean $(2n+1)$-space. Sci. Record (N.S.) 2 1958 271--275.

In it he proves any two $C^r$-embeddings $M^n \to \mathbb R^{2n+1}$ are isotopic, provided $n>1$. $M^n$ is any compact $n$-manifold (the title does not mention he also assumed connected, but it is). The techniques are now standard -- your two embeddings $f_0, f_1 : M \to \mathbb R^{2n+1}$ are homotopic $f_t : M \to \mathbb R^{2n+1}$ so you look at the "graph", $(x,t) \longmapsto (f_t(x), t)$ as a map $M \times [0,1] \to \mathbb R^{2n+!} \times [0,1]$ and then approximate by an immersion and consider appropriate usage of the Whitney trick.

I always thought that Haefliger's result applied in dimensions higher than 5, but in skimming the manuscript, it looks like 5 might be a critical case. Please read at page 404 and 405 carefully.

To address your original question, that surfaces in 4-space can still intersect and that they can be knotted may not effect the potential knottedness in dimension 5. On the other hand, it is easy to construct knottings of 3-manifolds in 5-space by spinning and twist-spinning.