Advanced Studies in Pure Mathematics

Cobordism of fibered knots and related topics

Vincent Blanlœil and Osamu Saeki

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Abstract

This is a survey article on the cobordism theory of non-spherical knots studied in [BM, B2, BS1, BMS, BS2, BS3]. Special emphasis is put on fibered knots.

We first recall the classical results concerning cobordisms of spherical knots. Then we give recent results on cobordisms of simple fibered $(2n-1)$-knots for $n \ge 2$ together with relevant examples. We discuss the Fox-Milnor type relation and show that the usual spherical knot cobordism group modulo the subgroup generated by the cobordism classes of fibered knots is infinitely generated for odd dimensions. The pull back relation on the set of knots is also discussed, which is closely related to the cobordism theory of knots via the codimension two surgery theory. We also present recent results on cobordisms of surface knots in $S^4$ and 4-dimensional knots in $S^6$. Finally we give some open problems related to the subject.

Article information

Source
Singularities in Geometry and Topology 2004, J.-P. Brasselet and T. Suwa, eds. (Tokyo: Mathematical Society of Japan, 2007), 1-47

Dates
Received: 7 April 2005
Revised: 30 December 2005
First available in Project Euclid: 16 December 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1544999903

Digital Object Identifier
doi:10.2969/aspm/04610001

Mathematical Reviews number (MathSciNet)
MR2342884

Zentralblatt MATH identifier
1128.57024

Subjects
Primary: 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}
Secondary: 57Q60: Cobordism and concordance 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45] 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]

Keywords
Fibered knot knot cobordism algebraic knot complex hypersurface singularity algebraic cobordism Seifert form embedded surgery Spin structure

Citation

Blanlœil, Vincent; Saeki, Osamu. Cobordism of fibered knots and related topics. Singularities in Geometry and Topology 2004, 1--47, Mathematical Society of Japan, Tokyo, Japan, 2007. doi:10.2969/aspm/04610001. https://projecteuclid.org/euclid.aspm/1544999903


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