TY - JOUR

T1 - Classification of knotted tori in 2-metastable dimension

AU - Cencelj, Matija

AU - Repovš, Dušan

AU - Skopenkov, Mikhail

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The first and second authors were supported in part by the Slovenian Research Agency (grant nos. P1-0292-0101 and J1-4144-0101). The third author was supported in part by the Russian Foundation for Basic Research (grant no. 12-01-00748-a), the Programme of the President of the Russian Federation for the Support of Young Scientists (grant no. MK-3965.2012.1), the "Dynasty" Foundation and the Simons Foundation.

PY - 2013/1/22

Y1 - 2013/1/22

N2 - This paper is devoted to the classical Knotting Problem: for a given manifold N and number m describe the set of isotopy classes of embeddings N → Sm. We study the specific case of knotted tori, that is, the embeddings Sp × Sq → Sm. The classification of knotted tori up to isotopy in the metastable dimension range m > p + 3 2 q + 2, p 6 q, was given by Haefliger, Zeeman and A. Skopenkov. We consider the dimensions below the metastable range and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that p+ 4 3 q +2 < mp+ 3 2 q +2 and m > 2p+q +2. Then the set of isotopy classes of smooth embeddings Sp × Sq → Sm is infinite if and only if either q + 1 or p + q + 1 is divisible by 4. © 2012 RAS(DoM) and LMS.

AB - This paper is devoted to the classical Knotting Problem: for a given manifold N and number m describe the set of isotopy classes of embeddings N → Sm. We study the specific case of knotted tori, that is, the embeddings Sp × Sq → Sm. The classification of knotted tori up to isotopy in the metastable dimension range m > p + 3 2 q + 2, p 6 q, was given by Haefliger, Zeeman and A. Skopenkov. We consider the dimensions below the metastable range and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that p+ 4 3 q +2 < mp+ 3 2 q +2 and m > 2p+q +2. Then the set of isotopy classes of smooth embeddings Sp × Sq → Sm is infinite if and only if either q + 1 or p + q + 1 is divisible by 4. © 2012 RAS(DoM) and LMS.

UR - http://hdl.handle.net/10754/562427

UR - http://arxiv.org/abs/arXiv:0811.2745v3

UR - http://www.scopus.com/inward/record.url?scp=84873858760&partnerID=8YFLogxK

U2 - 10.1070/SM2012v203n11ABEH004281

DO - 10.1070/SM2012v203n11ABEH004281

M3 - Article

SN - 1064-5616

VL - 203

SP - 1654

EP - 1681

JO - Sbornik: Mathematics

JF - Sbornik: Mathematics

IS - 11

ER -