Embeddings in Euclidean space: plan and convention

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1 Introduction

This page gives references to pages on the classification of embeddings, and introduces notation and conventions used there.

2 References to pages on the classification of embeddings

Here is the introductory article:

• Embeddings in Euclidean space: an introduction to their classification [Skopenkov2016c]

Below we list references to information about the classification of embeddings of manifolds into Euclidean space.

The first list is structured by the dimension of the source manifold and the target Euclidean space:

• Embeddings just below the stable range: classification [Skopenkov2016e]

• 3-manifolds in 6-space [Skopenkov2016t]

• 4-manifolds in 7-space [Skopenkov2016f]

Information structured by the `complexity' of the source manifold:

• Knots, i.e. embeddings of spheres

• High codimension links [Skopenkov2016h]

• Knotted tori [Skopenkov2016k]

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

3 Notation and conventions

The following notations and conventions will be used in some other pages about embeddings, including those listed in $\S$${{Authors|Askopenkov}} == Introduction == <wikitex>; \label{s:intr} This page gives references to pages on the classification of embeddings, and introduces notation and conventions used there. </wikitex> == References to pages on the classification of embeddings == <wikitex>; \label{s:list} Here is the introductory article: * [[Embeddings in Euclidean space: an introduction to their classification]] \cite{Skopenkov2016c} Below we list references to information about the classification of embeddings of manifolds into Euclidean space. The first list is structured by the dimension of the source manifold and the target Euclidean space: * [[Embeddings just below the stable range: classification]] \cite{Skopenkov2016e} * [[3-manifolds in 6-space]] \cite{Skopenkov2016t} * [[4-manifolds in 7-space]] \cite{Skopenkov2016f} Information structured by the `complexity' of the source manifold: * [[Knots, i.e. embeddings of spheres]] * [[High codimension links]] \cite{Skopenkov2016h} * [[Knotted tori]] \cite{Skopenkov2016k} For more information see e.g. \cite{Skopenkov2006}. </wikitex> <!--[[Unknotting of projective spaces]] [[Embeddings of highly-connected manifolds: classification]] [[Link maps: classification]] [[Immersions: classification]] [[Embeddings of manifolds with boundary]] [[Some open problems]] --> == Notation and conventions == <wikitex>; \label{s:nc} The following notations and conventions will be used in some other pages about embeddings, including those listed in $\S$\ref{s:list}. For a manifold $N$ let $E^m_D(N)$ or $E^m_{PL}(N)$ denote the set of [[Wikipedia:Differential_manifold#Differentiable_functions|smooth]] or [[Wikipedia:Piecewise_linear_function|piecewise-linear]] (PL) [[Embedding|embeddings]] $N\to S^m$ up to smooth or PL [[isotopy]]. If the category is omitted, then the result holds (or a definition or a construction is given) in both categories. The sources of all embeddings are 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 $\varepsilon(k):=1-(-1)^k$ be <script type="math/tex">\S$2.

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

The sources of all embeddings are 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 $\varepsilon(k):=1-(-1)^k$$\varepsilon(k):=1-(-1)^k$ be $0$$0$ for $k$$k$ even and $2$$2$ for $k$$k$ odd, so that $\Zz_{\varepsilon(k)}$$\Zz_{\varepsilon(k)}$ is $\Zz$$\Zz$ for $k$$k$ even and $\Zz_2$$\Zz_2$ for $k$$k$ odd.

Denote by $V_{m,n}$$V_{m,n}$ the Stiefel manifold of orthonormal $n$$n$-frames in $\Rr^m$$\Rr^m$.

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

For a manifold $P$$P$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$H_s(P,\partial):=H_s(P,\partial P)$.

A closed manifold $N$$N$ is called homologically $k$$k$-connected, if $N$$N$ is connected and $H_i(N)=0$$H_i(N)=0$ for every $i=1,\dots,k$$i=1,\dots,k$. This condition is equivalent to $\tilde H_i(N)=0$$\tilde H_i(N)=0$ for each $i=0,\dots,k$$i=0,\dots,k$, where $\tilde H_i$$\tilde H_i$ are reduced homology groups. A pair $(N,\partial N)$$(N,\partial N)$ is called homologically $k$$k$-connected, if $H_i(N,\partial)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$i=0,\dots,k$.

The self-intersection set of a map $f:X\to Y$$f:X\to Y$ is $\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$$\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$

For a smooth 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 tight enough 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 linear normal bundle of $f$$f$ to the subspace of unit length vectors identified with $\partial C_f$$\partial C_f$.
• $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$$\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ and $A_f:H_s(N)\to H_{s+1}(C,\partial)$$A_f:H_s(N)\to H_{s+1}(C,\partial)$ the homological Alexander duality isomorphisms, see the well-known Alexander Duality Lemmas of [Skopenkov2008], [Skopenkov2005].

4 References

• [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
• [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.
• [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
• [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.

$ for $k$ even and $ for $k$ odd, so that $\Zz_{\varepsilon(k)}$ is $\Zz$ for $k$ even and $\Zz_2$ for $k$ odd. Denote by $V_{m,n}$ the Stiefel manifold of orthonormal $n$-frames in $\Rr^m$. We omit $\Zz$-coefficients from the notation of (co)homology groups. For a manifold $P$ with boundary $\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$. A closed manifold $N$ is called ''homologically $k$-connected'', if $N$ is connected and $H_i(N)=0$ for every $i=1,\dots,k$. This condition is equivalent to $\tilde H_i(N)=0$ for each $i=0,\dots,k$, where $\tilde H_i$ are reduced homology groups. A pair $(N,\partial N)$ is called ''homologically $k$-connected'', if $H_i(N,\partial)=0$ for every $i=0,\dots,k$. The ''self-intersection set'' of a map $f:X\to Y$ is $\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$ For a smooth embedding $f:N\to\Rr^m$ denote by * $C_f$ the closure of the complement in $S^m\supset\Rr^m$ to a tight enough tubular neighborhood of $f(N)$ and * $\nu_f:\partial C_f\to N$ the restriction of the linear normal bundle of $f$ to the subspace of unit length vectors identified with $\partial C_f$. * $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ and $A_f:H_s(N)\to H_{s+1}(C,\partial)$ the homological Alexander duality isomorphisms, see the well-known Alexander Duality Lemmas of \cite{Skopenkov2008}, \cite{Skopenkov2005}. == References == {{#RefList:}} [[Category:Theory]] [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S2.

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

The sources of all embeddings are 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 $\varepsilon(k):=1-(-1)^k$$\varepsilon(k):=1-(-1)^k$ be $0$$0$ for $k$$k$ even and $2$$2$ for $k$$k$ odd, so that $\Zz_{\varepsilon(k)}$$\Zz_{\varepsilon(k)}$ is $\Zz$$\Zz$ for $k$$k$ even and $\Zz_2$$\Zz_2$ for $k$$k$ odd.

Denote by $V_{m,n}$$V_{m,n}$ the Stiefel manifold of orthonormal $n$$n$-frames in $\Rr^m$$\Rr^m$.

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

For a manifold $P$$P$ with boundary $\partial P$$\partial P$ denote $H_s(P,\partial):=H_s(P,\partial P)$$H_s(P,\partial):=H_s(P,\partial P)$.

A closed manifold $N$$N$ is called homologically $k$$k$-connected, if $N$$N$ is connected and $H_i(N)=0$$H_i(N)=0$ for every $i=1,\dots,k$$i=1,\dots,k$. This condition is equivalent to $\tilde H_i(N)=0$$\tilde H_i(N)=0$ for each $i=0,\dots,k$$i=0,\dots,k$, where $\tilde H_i$$\tilde H_i$ are reduced homology groups. A pair $(N,\partial N)$$(N,\partial N)$ is called homologically $k$$k$-connected, if $H_i(N,\partial)=0$$H_i(N,\partial)=0$ for every $i=0,\dots,k$$i=0,\dots,k$.

The self-intersection set of a map $f:X\to Y$$f:X\to Y$ is $\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$$\Sigma(f):=\{x\in X\ :\ |f^{-1}fx|>1\}.$

For a smooth 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 tight enough 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 linear normal bundle of $f$$f$ to the subspace of unit length vectors identified with $\partial C_f$$\partial C_f$.
• $\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$$\widehat A_f:H_s(N)\to H_{s+m-n-1}(C)$ and $A_f:H_s(N)\to H_{s+1}(C,\partial)$$A_f:H_s(N)\to H_{s+1}(C,\partial)$ the homological Alexander duality isomorphisms, see the well-known Alexander Duality Lemmas of [Skopenkov2008], [Skopenkov2005].

4 References

• [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
• [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.
• [Skopenkov2008] A. Skopenkov, A classification of smooth embeddings of 3-manifolds in 6-space, Math. Z. 260 (2008), no.3, 647–672. Available at the arXiv:0603429MR2434474 (2010e:57028) Zbl 1167.57013

• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.
• [Skopenkov2016e] A. Skopenkov, Embeddings just below the stable range: classification, to appear in Bull. Man. Atl.
• [Skopenkov2016f] A. Skopenkov, 4-manifolds in 7-space, to appear in Bull. Man. Atl.
• [Skopenkov2016h] A. Skopenkov, High codimension links, to appear in Bull. Man. Atl.
• [Skopenkov2016k] A. Skopenkov, Knotted tori, preprint.
• [Skopenkov2016t] A. Skopenkov, 3-manifolds in 6-space, to appear in Boll. Man. Atl.