Niveau: Supérieur, Doctorat, Bac+8
Fibered knots in low and high dimensions Vincent Blanlœil Abstract. This paper is a survey of cobordism theory for knots. We first recall the classical results of knot cobordism, and next we give some recent results. We classify fibered knot up to cobordism in all dimensions, and we give several examples of fibered knots which are cobordant and non isotopic. Some related results about surfaces embedded in S4 are given. ”... the theory of ”Cobordisme” which has, within the few years of its existence, led to the most penetrating insights into the topology of differentiable manifolds.” H. Hopf, International Congress of Mathematics, 1958. 1. Introduction 1.1. Historic. In the sixties, R. Fox and J. Milnor [F-M] were the first to study cobordism of embeddings of S1 in S3. Next, M. Kervaire [K2] and J. Levine [L2] studied embeddings of (2n ? 1)-spheres into codimension two spheres, and gave a classification of these embeddings up to cobordism. Moreover, M. Kervaire defined a group structure on the sets C2n?1, of cobordism classes of (2n?1)-spheres embedded in S2n+1, and C˜2n?1, of concordance classes of (2n?1)-spheres embedded in S2n+1. Remark that spherical knots were only studied as codimension two embeddings into spheres, because in the P.
- manifolds into
- dimensional knots
- knots associated
- seifert manifold
- cobordant knots
- knots isotopy
- study cobordism
- well known since
- connected manifold
- spherical knots