LINK COBORDISM AND THE INTERSECTION OF SLICE DISCS

Abstract

We work in the smooth category.

An (oriented) (ordered) m-component n-(dimensional) link is a smooth oriented submanifold L = {K₁, …, K_m} of Sⁿ⁺² which is the ordered disjoint union of m manifolds, each PL-homeomorphic to the standard n-sphere. If m = 1, then L is called a knot.

We say that m-component n-dimensional links L₀ and L₁ are (link-)concordant or (link-)cobordant if there is a smooth oriented submanifold C˜ = {C₁, …, C_m} of Sⁿ⁺² × [0, 1] which meets the boundary transversely in ∂C˜, is PL-homeomorphic to L₀ × [0, 1], and meets Sⁿ⁺² × {l} in L_l (l = 0, 1). If m = 1, then we say that n-knots L₀ and L_l are (knot-)concordant or (knot-)cobordant. Then we call C a concordance-cylinder of the two n-knots L₀ and L_l.

If an n-link L is concordant to the trivial link, then we call L a slice link.

If an n-link L = {K₁, …, K_m} ⊂ Sⁿ⁺² = ∂Bⁿ⁺³ ⊂ Bⁿ⁺³ is slice, then there is a disjoint union of (n + 1)-discs Dⁿ⁺¹₁ [amalg ] … [amalg ] Dⁿ⁺¹_m in Bⁿ⁺³ such that Dⁿ⁺¹_i ∩ Sⁿ⁺² = ∂Dⁿ⁺¹_i = K_i. (Dⁿ⁺¹₁, …, Dⁿ⁺¹_m) is called a set of slice discs for L. If m = 1, then Dⁿ⁺¹₁ is called a slice disc for the knot L.