Cobordism of fibered knots and related topics

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.