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 and restrictions

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

• $The Homeomorphism Problem:$$== 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:$$The Embedding Problem:$ When does a given space embed into $\Rr^m$$\Rr^m$?
• $The Knotting Problem:$$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}}$${\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 knot theory, %\linebreak ${\underline{complete\ classification\ of\ links\ by\ M.Skopenkov}}$${\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$$N$ let $E^m_D(N)$$E^m_D(N)$ or $E^m_{PL}(N)$$E^m_{PL}(N)$ denote the set of smooth or PL embeddings $N\to\R^m$$N\to\R^m$ up to smooth or PL isotopy. %The sign $\sim_{PL}$$\sim_{PL}$ or $\sim_D$$\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$$B^n$ be a closed $n$$n$-ball in a closed connected $n$$n$-manifold $N$$N$.

Denote

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$N_0:=\Cl(N-B^n)$.

Let

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$\Z_{(k)}$ be $\Z$$\Z$ for $k$$k$ even and

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$\Z_2$ for $k$$k$ odd.

We omit $\Z$$\Z$-coefficients from the notation of (co)ho\-mo\-lo\-gy groups.

%From now on

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$f:N\to\R^{2n}$ is an embedding, unless another meaning of $f$$f$

%is explicitly given.

For an embedding

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$f:N\to\R^{2n}$ denote by

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$\bullet$ $C_f$$C_f$ the closure of the complement in

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$S^m\supset\R^m$ to a tubular

neighborhood of $f(N)$$f(N)$ and

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$\bullet$

$\nu_f:\partial C_f\to N$$\nu_f:\partial C_f\to N$ the restriction of the normal bundle of $f$$f$.

2 References

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