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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


We classify smooth Brunnian (i.e., unknotted on both components) embeddings (S2 × S1 ) ⊔ S3 → ℝ6. Any Brunnian embedding (S2 × S1 ) ⊔ S3 → ℝ6 is isotopic to an explicitly constructed embedding fk,m,n for some integers k, m, n such that mn (mod 2). Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if k = k′, mm′ (mod 2k) and nn′ (mod 2k). We use Haefliger’s classification of embeddings S3S3 → ℝ6 in our proof. The relation between the embeddings (S2 × S1 ) ⊔ S3 → ℝ6 and S3S3 → ℝ6 is not trivial, however. For example, we show that there exist embeddings f: (S2 × S1 ) ⊔ S3 → ℝ6 and g, g′: S3S3 → ℝ6 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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