# When is the set of embeddings finite up to isotopy

@article{Skopenkov2011WhenIT, title={When is the set of embeddings finite up to isotopy}, author={Mikhail Skopenkov}, journal={International Journal of Mathematics}, year={2011}, volume={26}, pages={1550051} }

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddingsN → Smfinite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when N is a product of two spheres. In the following theorem, FCS(i, j) ⊂ ℤ2 is a specific set depending only on the parity of…

## 3 Citations

### How do autodiffeomorphisms act on embeddings?

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2017

We work in the smooth category. The following problem was suggested by E. Rees in 2002: describe the precomposition action of self-diffeomorphisms of Sp × Sq on the set of isotopy classes of…

### Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2022

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; {\mathbb Z} )$. Our main result is a readily calculable…

### Classification of knotted tori

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2020

Abstract For a smooth manifold N denote by Em(N) the set of smooth isotopy classes of smooth embeddings N → ℝm. A description of the set Em(Sp × Sq) was known only for p = q = 0 or for p = 0, m ≠ q +…

## References

SHOWING 1-10 OF 26 REFERENCES

### A CLASSIFICATION OF SMOOTH EMBEDDINGS OF FOUR-MANIFOLDS IN SEVEN-SPACE, II

- Mathematics
- 2011

Let N be a closed connected smooth four-manifold with H1(N; ℤ) = 0. Our main result is the following classification of the set E7(N) of smooth embeddings N → ℝ7 up to smooth isotopy. Haefliger proved…

### A new invariant and parametric connected sum of embeddings

- Mathematics, Computer Science
- 2005

An isotopy invariant of embeddings N -> R^m of manifolds into Euclidean space is defined and parametric connected sum of certain embedDings (analogous to surgery) is defined to obtain new completeness results for the \alpha-invariant and the following estimation of isotopy classes ofembeddings.

### Embeddings from the point of view of immersion theory

- Philosophy
- 1999

Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M,N) should come from an analysis of the cofunctor V |-->…

### CLASSIFICATION OF EMBEDDINGS BELOW THE METASTABLE DIMENSION

- Mathematics
- 2006

It is developed a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of…

### On the Browder-Levine-Novikov Embedding Theorems

- Mathematics
- 2004

In this survey, we apply the concepts of complement and neighborhood to embed- dings of manifolds into Euclidean space (in codimension at least three). We describe how a combination of these concepts…

### The rational classification of links of codimension > 2

- Mathematics, Computer Science
- 2011

This work determines the rank of the group in general and determines precisely when is finite, based on the Haefliger exact sequence for groups of links and the theory of Lie algebras.

### Classification of knotted tori in 2-metastable dimension

- MathematicsSbornik: Mathematics
- 2012

This paper is devoted to the classical Knotting Problem: for a given manifold and number describe the set of isotopy classes of embeddings . We study the specific case of knotted tori, that is, the…

### A classification of smooth embeddings of 3-manifolds in 6-space

- Mathematics
- 2008

We work in the smooth category. If there are knotted embeddings $${S^n \to \mathbb{R}^m}$$, which often happens for 2m < 3n + 4, then no explicit complete description of the embeddings of n-manifolds…

### On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

- Mathematics
- 2002

Abstract. We prove beyond the metastable dimension the PL cases of the classical theorems due to Haefliger, Harris, Hirsch and Weber on the deleted product criteria for embeddings and immersions. The…