A theory of concordance for non-spherical 3-knots

Blanlœil, Vincent; Saeki, Osamu

doi:10.1090/S0002-9947-02-03024-6

A theory of concordance for non-spherical 3-knots
HTML articles powered by AMS MathViewer

by Vincent Blanlœil and Osamu Saeki PDF

Trans. Amer. Math. Soc. 354 (2002), 3955-3971 Request permission

Abstract:

Consider a closed connected oriented 3-manifold embedded in the $5$-sphere, which is called a $3$-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.

References

Vincent Blanlœil and Françoise Michel, A theory of cobordism for non-spherical links, Comment. Math. Helv. 72 (1997), no. 1, 30–51. MR 1456314, DOI 10.1007/PL00000365
Steven Boyer, Simply-connected $4$-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), no. 1, 331–357. MR 857447, DOI 10.1090/S0002-9947-1986-0857447-6
Alan H. Durfee, Fibered knots and algebraic singularities, Topology 13 (1974), 47–59. MR 336750, DOI 10.1016/0040-9383(74)90037-8
Jonathan A. Hillman, Simple locally flat $3$-knots, Bull. London Math. Soc. 16 (1984), no. 6, 599–602. MR 758132, DOI 10.1112/blms/16.6.599
Steve J. Kaplan, Constructing framed $4$-manifolds with given almost framed boundaries, Trans. Amer. Math. Soc. 254 (1979), 237–263. MR 539917, DOI 10.1090/S0002-9947-1979-0539917-X
Louis H. Kauffman, Branched coverings, open books and knot periodicity, Topology 13 (1974), 143–160. MR 375337, DOI 10.1016/0040-9383(74)90005-6
C. Kearton, Some nonfibred $3$-knots, Bull. London Math. Soc. 15 (1983), no. 4, 365–367. MR 703762, DOI 10.1112/blms/15.4.365
Michel A. Kervaire, Les nœuds de dimensions supérieures, Bull. Soc. Math. France 93 (1965), 225–271 (French). MR 189052, DOI 10.24033/bsmf.1624
Michel A. Kervaire, Knot cobordism in codimension two, Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 83–105. MR 0283786
Terry Lawson, Trivializing $5$-dimensional $h$-cobordisms by stabilization, Manuscripta Math. 29 (1979), no. 2-4, 305–321. MR 545047, DOI 10.1007/BF01303633
J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR 246314, DOI 10.1007/BF02564525
John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372, DOI 10.1007/978-3-642-88330-9
U. Pinkall, Regular homotopy classes of immersed surfaces, Topology 24 (1985), no. 4, 421–434. MR 816523, DOI 10.1016/0040-9383(85)90013-8
Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
Osamu Saeki, On simple fibered knots in $S^5$ and the existence of decomposable algebraic $3$-knots, Comment. Math. Helv. 62 (1987), no. 4, 587–601. MR 920059, DOI 10.1007/BF02564464
Osamu Saeki, Knotted homology $3$-spheres in $S^5$, J. Math. Soc. Japan 40 (1988), no. 1, 65–75. MR 917395, DOI 10.2969/jmsj/04010065
Osamu Saeki, Cobordism classification of knotted homology $3$-spheres in $S^5$, Osaka J. Math. 25 (1988), no. 1, 213–222. MR 937196
Osamu Saeki, Theory of fibered 3-knots in $S^5$ and its applications, J. Math. Sci. Univ. Tokyo 6 (1999), no. 4, 691–756. MR 1742599
Osamu Saeki, On punctured $3$-manifolds in $5$-sphere, Hiroshima Math. J. 29 (1999), no. 2, 255–272. MR 1704247
O. Saeki, A. Szűcz, and M. Takase, Regular homotopy classes of immersions of $3$-manifolds into $5$-space, preprint, 2000.
C. T. C. Wall, Diffeomorphisms of $4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR 163323, DOI 10.1112/jlms/s1-39.1.131