# Moscow Mathematical Journal

Volume 16, Issue 1, January–March 2016 pp. 1–25.

The Classification of Certain Linked 3-Manifolds in 6-Space

**Authors**:
S. Avvakumov

**Author institution:** Institute of Science and Technology Austria, IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria

**Summary: **

We classify smooth Brunnian (i.e., unknotted on both components) embeddings (*S*^{2} × *S*^{1} ) ⊔ *S*^{3} → ℝ^{6}. Any Brunnian embedding
(*S*^{2} × *S*^{1} ) ⊔ *S*^{3} → ℝ^{6} is isotopic to an explicitly constructed embedding
*f _{k,m,n}* for some integers

*k, m, n*such that

*m*≡

*n*(mod 2). Two embeddings

*f*and

_{k,m,n}*f*are isotopic if and only if

_{k′,m′,n′}*k*=

*k*′,

*m*≡

*m*′ (mod 2

*k*) and

*n*≡

*n*′ (mod 2

*k*). We use Haefliger’s classification of embeddings

*S*

^{3}⊔

*S*

^{3}→ ℝ

^{6}in our proof. The relation between the embeddings (

*S*

^{2}×

*S*

^{1}) ⊔

*S*

^{3}→ ℝ

^{6}and

*S*

^{3}⊔

*S*

^{3}→ ℝ

^{6}is not trivial, however. For example, we show that there exist embeddings

*f*: (

*S*

^{2}×

*S*

^{1}) ⊔

*S*

^{3}→ ℝ

^{6}and

*g, g*′:

*S*

^{3}⊔

*S*

^{3}→ ℝ

^{6}such that the componentwise embedded connected sum

*f*#

*g*is isotopic to

*f*#

*g*′ but

*g*is not isotopic to

*g*′.

2010 Math. Subj. Class. Primary: 57R40, 57R52; Secondary: 57Q45, 55P10

**Keywords:**Classification of embeddings, framed cobordism, linked manifolds.

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