𝑘-cobordism for links in 𝑆³

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We give an explicit finite set of (based) links which generates, under connected sum, the k k -cobordism classes of links. We show that the union of these generating sets, 2 ≤ k > ∞ 2 \leq k > \infty , is not a generating set for ω \omega -cobordism classes or even ∞ \infty -cobordism classes. For 2 2 -component links in S 3 {S^3} we define ( 2 , k ) (2,k) -corbordism and show that the concordance invariants β i , i ∈ Z + {\beta ^i},i \in {\mathbb {Z}^+} , previously defined by the author, are invariants under ( 2 , i + 1 ) (2,i + 1) -cobordism. Moreover we show that the ( 2 , k ) (2,k) -cobordism classes of links (with linking number 0) is a free abelian group of rank k − 1 k - 1 , detected precisely by β 1 × ⋯ × β k − 1 {\beta ^1} \times \cdots \times {\beta ^{k - 1}} . We write down a basis. The union of these bases ( 2 ≤ k > ∞ ) (2 \leq k > \infty ) is not a generating set for ( 2 , ∞ ) (2,\infty ) or ( 2 , ω ) (2,\omega ) -cobordism classes. However, we can show that ∏ i = 1 ∞ β i ( ) \prod _{i = 1}^\infty {\beta ^i}(\;) is an isomorphism from the group of ( 2 , ∞ ) (2,\infty ) -cobordism classes to the subgroup R ⊂ ∏ i = 1 ∞ Z \mathcal {R} \subset \prod _{i = 1}^\infty \mathbb {Z} of linearly recurrent sequences, so a basis exists by work of T. Jin.