# Embeddings in Euclidean space: plan and convention

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## Contents |

## 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:

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:

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

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 2.

For a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .

Let be for even and for odd, so that is for even and for odd.

Denote by the Stiefel manifold of orthonormal -frames in .

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

For a manifold with boundary denote .

A closed manifold is called *homologically -connected*, if is connected and for every .
This condition is equivalent to for each , where are reduced homology groups.
A pair is called *homologically -connected*, if for every .

The *self-intersection set* of a map is

For a smooth embedding denote by

- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
- and 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 a manifold let or denote the set of smooth or piecewise-linear (PL) embeddings 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 be a closed -ball in a closed connected -manifold . Denote .

Let be for even and for odd, so that is for even and for odd.

Denote by the Stiefel manifold of orthonormal -frames in .

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

For a manifold with boundary denote .

A closed manifold is called *homologically -connected*, if is connected and for every .
This condition is equivalent to for each , where are reduced homology groups.
A pair is called *homologically -connected*, if for every .

The *self-intersection set* of a map is

For a smooth embedding denote by

- the closure of the complement in to a tight enough tubular neighborhood of and
- the restriction of the linear normal bundle of to the subspace of unit length vectors identified with .
- and 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.