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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Locally flat $ 2$-knots in $ S\sp 2\times S\sp 2$ with the same fundamental group


Author: Yoshihisa Sato
Journal: Trans. Amer. Math. Soc. 323 (1991), 911-920
MSC: Primary 57Q45
DOI: https://doi.org/10.1090/S0002-9947-1991-0986701-4
MathSciNet review: 986701
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Abstract: We consider a locally flat $ 2$-sphere in $ {S^2} \times {S^2}$ representing a primitive homology class $ \xi $, which is referred to as a $ 2$-knot in $ {S^2} \times {S^2}$ representing $ \xi $. Then for any given primitive class $ \xi $, there exists a $ 2$-knot in $ {S^2} \times {S^2}$ representing $ \xi $ with simply-connected complement. In this paper, we consider the classification of $ 2$-knots in $ {S^2} \times {S^2}$ whose complements have a fixed fundamental group. We show that if the complement of a $ 2$-knot $ S$ in $ {S^2} \times {S^2}$ is simply connected, then the ambient isotopy type of $ S$ is determined. In the case of nontrivial $ {\pi _1}$, however, we show that the ambient isotopy type of a $ 2$-knot in $ {S^2} \times {S^2}$ with nontrivial $ {\pi _1}$ is not always determined by $ {\pi _1}$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-0986701-4
Keywords: $ 2$-knot, $ {S^2} \times {S^2}$, homology $ 3$-sphere
Article copyright: © Copyright 1991 American Mathematical Society

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