Embeddings in Euclidean space: an introduction to their classification

This page has been accepted for publication in the Bulletin of the Manifold Atlas.

1 Introduction

According to Zeeman, the classical problems of topology are the following.

• $The$$== Introduction == <wikitex>; According to Zeeman, the classical problems of topology are the following. * $The$ $Homeomorphism$ $Problem:$ When are two given spaces homeomorphic? * $The$ $Embedding$ $Problem:$ When does a given space embed into $\Rr^m$? * $The$ $Knotting$ $Problem:$ When are two given embeddings isotopic? This article concerns the Knotting Problem. We recall all known $complete$ $readily$ $calculable$ [[isotopy]] classification results for [[embeddings]] of $closed$ $connected$ manifolds into Euclidean spaces. This is done in particular for codimension 1 and 2 embeddings. (There is an extensive study of codimension 2 embeddings not directly aiming at complete classification. Almost none is said here about that. So in the title of this article we only mention embeddings of codimension greater than 2. See more in [[Wikipedia:Knot_theory|knot theory]].) </wikitex> == Notation and conventions used in the links below == <wikitex>; For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of [[smooth]] or [[piecewise-linear]] (PL) embeddings $N\to\Rr^m$ up to smooth or PL isotopy. If a category is omitted, then the result holds (or a definition or a construction is given) in both categories. All manifolds are tacitly assumed to be compact. Let $B^n$ be a closed $n$-ball in a closed connected $n$-manifold $N$. Denote $N_0:=Cl(N-B^n)$. Let $\Zz_{(k)}$ be $\Zz$ for $k$ even and $\Zz_2$ for $k$ odd. We omit $\Zz$-coefficients from the notation of (co)homology groups. For an embedding $f:N\to\Rr^m$ denote by * $C_f$ the closure of the complement in $S^m\supset\Rr^m$ to a tubular neighborhood of $f(N)$ and *$\nu_f:\partial C_f\to N$ the restriction of the spherical normal bundle of $f$. </wikitex> == Links to specific results and examples == <wikitex>; [[Some general remarks]] [[Hudson tori|Example of Hudson tori]] [[Knots, i.e. embeddings of spheres]] [[Classification just below the stable range]] [[Examples of embeddings of 3-manifolds into the 6-space]] [[Classification of embeddings of 3-manifolds in the 6-space]] [[Definition of the Kreck invariant for 3-manifolds in the 6-space]] [[Classification for m=2n-1]] [[Examples of embeddings of 4-manifolds into the 7-space]] [[Classification of embeddings of 4-manifolds into the 7-space]] [[Links, i.e. embeddings of non-connected manifolds]] [[Embeddings of manifolds with boundary]] [[Some open problems]] For more information see e.g. \cite{Skopenkov2006}. </wikitex> == References == <wikitex>; {{#RefList:}} </wikitex> [[Category:Manifolds]] {{Stub}}The$ $Homeomorphism$$Homeomorphism$ $Problem:$$Problem:$ When are two given spaces homeomorphic?
• $The$$The$ $Embedding$$Embedding$ $Problem:$$Problem:$ When does a given space embed into $\Rr^m$$\Rr^m$?
• $The$$The$ $Knotting$$Knotting$ $Problem:$$Problem:$ When are two given embeddings isotopic?

This article concerns the Knotting Problem. We recall all known $complete$$complete$ $readily$$readily$ $calculable$$calculable$ isotopy classification results for embeddings of $closed$$closed$ $connected$$connected$ manifolds into Euclidean spaces. This is done in particular for codimension 1 and 2 embeddings.

(There is an extensive study of codimension 2 embeddings not directly aiming at complete classification. Almost none is said here about that. So in the title of this article we only mention embeddings of codimension greater than 2. See more in knot theory.)

2 Notation and conventions used in the links below

For a manifold $N$$N$ let $E^m_D(N)$$E^m_D(N)$ or $E^m_{PL}(N)$$E^m_{PL}(N)$ denote the set of smooth or piecewise-linear (PL) embeddings $N\to\Rr^m$$N\to\Rr^m$ up to smooth or PL isotopy. If a category is omitted, then the result holds (or a definition or a construction is given) in both categories.

All manifolds are tacitly assumed to be compact.

Let $B^n$$B^n$ be a closed $n$$n$-ball in a closed connected $n$$n$-manifold $N$$N$. Denote $N_0:=Cl(N-B^n)$$N_0:=Cl(N-B^n)$.

Let $\Zz_{(k)}$$\Zz_{(k)}$ be $\Zz$$\Zz$ for $k$$k$ even and $\Zz_2$$\Zz_2$ for $k$$k$ odd.

We omit $\Zz$$\Zz$-coefficients from the notation of (co)homology groups.

For an embedding $f:N\to\Rr^m$$f:N\to\Rr^m$ denote by

• $C_f$$C_f$ the closure of the complement in $S^m\supset\Rr^m$$S^m\supset\Rr^m$ to a tubular neighborhood of $f(N)$$f(N)$ and
• $\nu_f:\partial C_f\to N$$\nu_f:\partial C_f\to N$ the restriction of the spherical normal bundle of $f$$f$.

3 Links to specific results and examples

Some general remarks

Example of Hudson tori

Knots, i.e. embeddings of spheres

Classification just below the stable range

Examples of embeddings of 3-manifolds into the 6-space

Classification of embeddings of 3-manifolds in the 6-space

Definition of the Kreck invariant for 3-manifolds in the 6-space

Classification for m=2n-1

Examples of embeddings of 4-manifolds into the 7-space

Classification of embeddings of 4-manifolds into the 7-space

Links, i.e. embeddings of non-connected manifolds

Embeddings of manifolds with boundary

Some open problems

For more information see e.g. [Skopenkov2006].

4 References

• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.

This page has not been refereed. The information given here might be incomplete or provisional.