Embeddings in Euclidean space: an introduction to their classification

Revision as of 13:48, 14 February 2010

1 Introduction

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

• $== Introduction and restrictions == <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. (Thus for 1- and 2- dimensional manifolds we only indicate that such results are not available.) We present constructions of embeddings and invariants. See more in [[Wikipedia:Knot_theory|knot theory]] <!--%\linebreak ${\underline{complete\ classification\ of\ links\ by\ M.Skopenkov}}$ and open problems below. Later we hope to add information for manifolds with boundary. For more information see --> and \cite{Skopenkov2006}. == Notation and conventions == For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of [[smooth]] or [[piecesise-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 in this note 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 $f:N\to\Rr^{2n}$ 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 normal bundle of $f$. </wikitex> == Links to specific results == <wikitex>; [[Hudson tori|Hudson tori]] </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}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 knot theory.)

In the links below we present constructions of embeddings and invariants. For more information see [Skopenkov2006].

Notation and conventions for the links below

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 normal bundle of $f$.

2 Links to specific results and examples

Example of Hudson tori

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

Embeddings of manifolds with boundary

3 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.