A theory of concordance for nonspherical 3knots
Authors:
Vincent Blanloeil and Osamu Saeki
Journal:
Trans. Amer. Math. Soc. 354 (2002), 39553971
MSC (2000):
Primary 57Q45; Secondary 57R40
Published electronically:
May 21, 2002
MathSciNet review:
1926861
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: Consider a closed connected oriented 3manifold embedded in the sphere, which is called a knot in this paper. For two such knots, we say that their Seifert forms are spin concordant, if they are algebraically concordant with respect to a diffeomorphism between the 3manifolds which preserves their spin structures. Then we show that two simple fibered 3knots are geometrically concordant if and only if they have spin concordant Seifert forms, provided that they have torsion free first homology groups. Some related results are also obtained.
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 V. Blanlil and F. Michel,
A theory of cobordism for nonspherical links, Comment. Math. Helv. 72 (1997), 3051. MR 98h:57049
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 S. Boyer,
Simplyconnected manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), 331357. MR 88b:57023
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 A. Durfee,
Fibered knots and algebraic singularities, Topology 13 (1974), 4759. MR 49:1523
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Simple locally flat knots, Bull. London Math. Soc. 16 (1984), 599602. MR 86a:57019
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 S. Kaplan,
Constructing framed manifolds with given almost framed boundaries, Trans. Amer. Math. Soc. 254 (1979), 237263. MR 82h:57015
 6.
 L. Kauffman,
Branched coverings, open books and knot periodicity, Topology 13 (1974), 143160. MR 51:11532
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 C. Kearton,
Some nonfibred knots, Bull. London Math. Soc. 15 (1983), 365367. MR 84m:57014
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 M. Kervaire,
Les nuds de dimensions supérieures, Bull. Soc. Math. France 93 (1965), 225271. MR 32:6479
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Knot cobordism in codimension two, ManifoldsAmsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Math. 197 (1971), Springer, Berlin, pp. 83105. MR 44:1016
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Trivializing dimensional cobordisms by stabilization, Manuscripta Math. 29 (1979), 305321. MR 80i:57024
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 J. Levine,
Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229244. MR 39:7618
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Symmetric bilinear forms, Ergebnisse Math., Band 73, Springer, Berlin, Heidelberg, New York, 1973. MR 58:22129
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Additional Information
Vincent Blanloeil
Affiliation:
Département de Mathématiques, Université Louis Pasteur Strasbourg I, 7 rue René Descartes, 67084 Strasbourg cedex, France
Email:
blanloeil@math.ustrasbg.fr
Osamu Saeki
Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki, Fukuova 8128581, Japan
Email:
saeki@math.kyushuu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994702030246
PII:
S 00029947(02)030246
Keywords:
Concordance,
3knot,
Seifert form,
algebraic concordance,
spin structure,
fibered knot
Received by editor(s):
May 12, 2001
Received by editor(s) in revised form:
February 15, 2002
Published electronically:
May 21, 2002
Additional Notes:
The second author has been supported in part by GrantinAid for Scientific Research (No. 11440022), Ministry of Education, Science and Culture, Japan, and was supported in part by Louis Pasteur University, France, during his stay there in September 2000.
Article copyright:
© Copyright 2002
American Mathematical Society
