Moscow Mathematical Journal
Volume 16, Issue 1, January–March 2016 pp. 1–25.
The Classification of Certain Linked 3-Manifolds in 6-Space
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 m ≡ n (mod 2). Two embeddings fk,m,n and fk′,m′,n′ are isotopic if and only if k = k′, m ≡ m′
(mod 2k) and n ≡ n′ (mod 2k).
We use Haefliger’s classification of embeddings S3 ⊔ S3 → ℝ6 in our
proof. The relation between the embeddings (S2 × S1 ) ⊔ S3 → ℝ6 and
S3 ⊔ S3 → ℝ6 is not trivial, however. For example, we show that there
exist embeddings f: (S2 × S1 ) ⊔ S3 → ℝ6 and g, g′: S3 ⊔ S3 → 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
Authors:
S. Avvakumov
Author institution: Institute of Science and Technology Austria, IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Summary:
Keywords: Classification of embeddings, framed cobordism, linked manifolds.
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