The classification of certain linked 3-manifolds in 6-space

S. Avvakumov, Moscow Mathematical Journal 16 (2016) 1–25.

Journal Article | Published | English

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Abstract
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′.
Publishing Year
Date Published
2016-01-01
Journal Title
Moscow Mathematical Journal
Acknowledgement
I thank A. Skopenkov for telling me about the problem and for his useful remarks. I also thank A. Sossinsky, A. Zhubr, M. Skopenkov, P. Akhmetiev, and an anonymous referee for their feedback. Author was partially supported by Dobrushin fellowship, 2013, and by RFBR grant 15-01-06302.
Volume
16
Issue
1
Page
1 - 25
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Cite this

Avvakumov S. The classification of certain linked 3-manifolds in 6-space. Moscow Mathematical Journal. 2016;16(1):1-25.
Avvakumov, S. (2016). The classification of certain linked 3-manifolds in 6-space. Moscow Mathematical Journal, 16(1), 1–25.
Avvakumov, Serhii. “The Classification of Certain Linked 3-Manifolds in 6-Space.” Moscow Mathematical Journal 16, no. 1 (2016): 1–25.
S. Avvakumov, “The classification of certain linked 3-manifolds in 6-space,” Moscow Mathematical Journal, vol. 16, no. 1, pp. 1–25, 2016.
Avvakumov S. 2016. The classification of certain linked 3-manifolds in 6-space. Moscow Mathematical Journal. 16(1), 1–25.
Avvakumov, Serhii. “The Classification of Certain Linked 3-Manifolds in 6-Space.” Moscow Mathematical Journal, vol. 16, no. 1, Independent University of Moscow, 2016, pp. 1–25.
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