A theory of concordance for non-spherical 3-knots

Authors:
Vincent Blanloeil and Osamu Saeki

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3955-3971

MSC (2000):
Primary 57Q45; Secondary 57R40

DOI:
https://doi.org/10.1090/S0002-9947-02-03024-6

Published electronically:
May 21, 2002

MathSciNet review:
1926861

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consider a closed connected oriented 3-manifold 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 3-manifolds which preserves their spin structures. Then we show that two simple fibered 3-knots 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.

**1.**V. Blanlil and F. Michel,*A theory of cobordism for non-spherical links*,

Comment. Math. Helv.**72**(1997), 30-51. MR**98h:57049****2.**S. Boyer,*Simply-connected -manifolds with a given boundary*,

Trans. Amer. Math. Soc.**298**(1986), 331-357. MR**88b:57023****3.**A. Durfee,*Fibered knots and algebraic singularities*,

Topology**13**(1974), 47-59. MR**49:1523****4.**J. A. Hillman,*Simple locally flat -knots*,

Bull. London Math. Soc.**16**(1984), 599-602. MR**86a:57019****5.**S. Kaplan,*Constructing framed -manifolds with given almost framed boundaries*,

Trans. Amer. Math. Soc.**254**(1979), 237-263. MR**82h:57015****6.**L. Kauffman,*Branched coverings, open books and knot periodicity*,

Topology**13**(1974), 143-160. MR**51:11532****7.**C. Kearton,*Some non-fibred -knots*,

Bull. London Math. Soc.**15**(1983), 365-367. MR**84m:57014****8.**M. Kervaire,*Les nuds de dimensions supérieures*,

Bull. Soc. Math. France**93**(1965), 225-271. MR**32:6479****9.**M. Kervaire,*Knot cobordism in codimension two*, Manifolds-Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Math.**197**(1971), Springer, Berlin, pp. 83-105. MR**44:1016****10.**T. Lawson,*Trivializing -dimensional -cobordisms by stabilization*,

Manuscripta Math.**29**(1979), 305-321. MR**80i:57024****11.**J. Levine,*Knot cobordism groups in codimension two*,

Comment. Math. Helv.**44**(1969), 229-244. MR**39:7618****12.**J. Milnor and D. Husemoller,*Symmetric bilinear forms*,

Ergebnisse Math., Band 73, Springer, Berlin, Heidelberg, New York, 1973. MR**58:22129****13.**U. Pinkall,*Regular homotopy classes of immersed surfaces*,

Topology**24**(1985), 421-434. MR**87e:57028****14.**F. Quinn,*The stable topology of -manifolds*,

Topology Appl.**15**(1983), 71-77. MR**84b:37023****15.**O. Saeki,*On simple fibered -knots in and the existence of decomposable algebraic -knots*,

Comment. Math. Helv.**62**(1987), 587-601. MR**88k:57030****16.**O. Saeki,*Knotted homology -spheres in*,

J. Math. Soc. Japan**40**(1988), 65-75. MR**89g:57032****17.**O. Saeki,*Cobordism classification of knotted homology -spheres in*,

Osaka J. Math.**25**(1988), 213-222. MR**89g:57033****18.**O. Saeki,*Theory of fibered -knots in and its applications*,

J. Math. Sci. Univ. Tokyo**6**(1999), 691-756. MR**2001b:57058****19.**O. Saeki,*On punctured -manifolds in -sphere*,

Hiroshima Math. J.**29**(1999), 255-272. MR**2000h:57045****20.**O. Saeki, A. Szucz, and M. Takase,*Regular homotopy classes of immersions of -manifolds into -space*,

preprint, 2000.**21.**C. T. C. Wall,*Diffeomorphisms of -manifolds*,

J. London Math. Soc.**39**(1964), 131-140. MR**29:626**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57Q45,
57R40

Retrieve articles in all journals with MSC (2000): 57Q45, 57R40

Additional Information

**Vincent Blanloeil**

Affiliation:
Département de Mathématiques, Université Louis Pasteur Strasbourg I, 7 rue René Descartes, 67084 Strasbourg cedex, France

Email:
blanloeil@math.u-strasbg.fr

**Osamu Saeki**

Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki, Fukuova 812-8581, Japan

Email:
saeki@math.kyushu-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-02-03024-6

Keywords:
Concordance,
3-knot,
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 Grant-in-Aid 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