Embeddings of manifolds with boundary: classification

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1 Introduction

Recall that some Unknotting Theorems hold for manifolds with boundary [Skopenkov2016c, $\S$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\S$3], [Skopenkov2006, $\S$$\S$2]. In this page we present results peculiar for manifold with non-empty boundary.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3].

If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

Theorem 1.1. Every $n$$n$-manifold $N$$N$ with nonempty boundary PL-embeds into $\R^{2n-1}$$\R^{2n-1}$.

This result can be found in [Horvatic1971, theorem 5.2]

2 Unknotting Theorems

Theorem 2.1. Assume $N$$N$ is a compact connected $n$$n$-manifold and either $m \ge 2n+2$$m \ge 2n+2$ or $m\ge2n+1\ge5$$m\ge2n+1\ge5$. Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Condition $m \ge 2n+2$$m \ge 2n+2$ stands for General Position Theorem and condition $m\ge2n+1\ge5$$m\ge2n+1\ge5$ stands for Whitney-Wu Unknotting Theorem, see Unknotting Theorems, theorems 2.1 and 2.2 respectivly.

Theorem 2.2. Assume $N$$N$ is a compact connected $n$$n$-manifold with non-empty boundary and one of the following conditions holds:

(a) $m \ge 2n\neq4$$m \ge 2n\neq4$

(b) $N$$N$ is $1$$1$-connected, $m \ge 2n - 1\ge3$$m \ge 2n - 1\ge3$

Then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Part (a) of this theorem in case $n>2$$n>2$ can be found in [Edwards1968, $\S$$\S$ 4, corollary 5]. Case $n=1$$n=1$ is clear. Case $n=2$$n=2$ can be derived from deleted square criterion, \site{Skopenkov2006}.

Theorem 2.3. [The Haefliger-Zeeman unknotting theorem] For every $n\ge2k + 2$$n\ge2k + 2$, $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and closed $k$$k$-connected $n$$n$-manifold $N$$N$, any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

Theorem 2.4. Assume $N$$N$ is a compact $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$. If $N$$N$ is $k$$k$-connected, $m\ge2n-k$$m\ge2n-k$ and $m-k>3$$m-k>3$ then any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

This result can be found in [Hudson1969, Theorem 10.3]

3 Construction and examples

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4 Invariants

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5 Classification

Theorem 5.1.[Becker-Glover] Let $N$$N$ be a closed homologically $k$$k$-connected $n$$n$-manifold and $m\ge 3n/2+2$$m\ge 3n/2+2$. The cone map $\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$$\Lambda: \mathrm{Emb}^m (N_0)\to\mathrm{Emb}^{m+1}(N)$ is one-to-one for $m\ge 2n-2k$$m\ge 2n-2k$ and is surjective for $m=2n-2k-1$$m=2n-2k-1$.

[Becker&Glover1971]

6 Further discussion

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7 References

• [Becker&Glover1971] J. Becker and H. Glover, Note on the embedding of manifolds in Euclidean space, Proc. Am. Math. Soc. 27 (1971), 405-410. MR0268903 (42 #3800) Zbl 0207.22402
• [Edwards1968] Edwards, C. H. Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR226629 Zbl 0167.52001
• [Horvatic1971] K. Horvatic, On embedding polyhedra and manifolds, Trans. Am. Math. Soc. 157 (1971), 417-436.
• [Hudson1969] J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR0248844 (40 #2094) Zbl 0189.54507
• [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.
• [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.