Embeddings in Euclidean space: an introduction to their classification

Revision as of 13:17, 12 February 2010

1 Introduction and restrictions

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 {\it complete readily calculable} [[isotopy]] classification results for ${\underline{embeddings}}$ of {\it 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 [[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 [Sk08]. \bigskip {\bf Notation and conventions.} For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of smooth or PL embeddings $N\to\R^m$ up to smooth or PL isotopy. %The sign $\sim_{PL}$ or $\sim_D$ between embeddings means that they are PL or %smoothly isotopic. 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 $\Z_{(k)}$ be $\Z$ for $k$ even and $\Z_2$ for $k$ odd. We omit $\Z$-coefficients from the notation of (co)ho\-mo\-lo\-gy groups. %From now on $f:N\to\R^{2n}$ is an embedding, unless another meaning of $f$ %is explicitly given. For an embedding $f:N\to\R^{2n}$ denote by $\bullet$ $C_f$ the closure of the complement in $S^m\supset\R^m$ to a tubular neighborhood of $f(N)$ and $\bullet$ $\nu_f:\partial C_f\to N$ the restriction of the normal bundle of $f$. </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. (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 knot theory and [Sk08].

1 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 $f$.

2 Links to specific results

Hudson torus

2 References