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Mosc. Math. J., 2016, Volume 16, Number 1, Pages 1–25 (Mi mmj592)  

This article is cited in 1 scientific paper (total in 1 paper)

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

S. Avvakumov

Institute of Science and Technology Austria, IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria

Abstract: We classify smooth Brunnian (i.e., unknotted on both components) embeddings $(S^2\times S^1)\sqcup S^3 \to\mathbb R^6$. Any Brunnian embedding $(S^2\times S^1)\sqcup S^3\to\mathbb R^6$ is isotopic to an explicitly constructed embedding $f_{k,m,n}$ for some integers $k,m,n$ such that $m\equiv n\pmod2$. Two embeddings $f_{k,m,n}$ and $f_{k',m',n'}$ are isotopic if and only if $k=k'$, $m\equiv m'\pmod{2k}$ and $n\equiv n'\pmod{2k}$.
We use Haefliger's classification of embeddings $S^3\sqcup S^3\to\mathbb R^6$ in our proof. The relation between the embeddings $(S^2\times S^1)\sqcup S^3\to\mathbb R^6$ and $S^3\sqcup S^3\to\mathbb R^6$ is not trivial, however. For example, we show that there exist embeddings $f\colon(S^2\times S^1)\sqcup S^3\to\mathbb R^6$ and $g,g'\colon S^3\sqcup S^3\to\mathbb R^6$ such that the componentwise embedded connected sum $f#g$ is isotopic to $f#g'$ but $g$ is not isotopic to $g'$.

Key words and phrases: classification of embeddings, framed cobordism, linked manifolds.

Funding Agency Grant Number
Dobrushin Foundation
Russian Foundation for Basic Research 15-01-06302
Supported in part by Dobrushin fellowship, 2013, and by RFBR grant 15-01-06302.


DOI: https://doi.org/10.17323/1609-4514-2016-16-1-1-25

Full text: http://www.mathjournals.org/.../2016-016-001-001.html
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Bibliographic databases:

MSC: Primary 57R40, 57R52; Secondary 57Q45, 55P10
Received: October 28, 2014; in revised form September 7, 2015
Language:

Citation: S. Avvakumov, “The classification of certain linked $3$-manifolds in $6$-space”, Mosc. Math. J., 16:1 (2016), 1–25

Citation in format AMSBIB
\Bibitem{Avv16}
\by S.~Avvakumov
\paper The classification of certain linked $3$-manifolds in $6$-space
\jour Mosc. Math.~J.
\yr 2016
\vol 16
\issue 1
\pages 1--25
\mathnet{http://mi.mathnet.ru/mmj592}
\crossref{https://doi.org/10.17323/1609-4514-2016-16-1-1-25}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3470574}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000386360200001}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Skopenkov, “How do autodiffeomorphisms act on embeddings?”, Proc. R. Soc. Edinb. Sect. A-Math., 148:4 (2018), 835–848  crossref  mathscinet  isi  scopus
  • Moscow Mathematical Journal
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