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Transactions of the American Mathematical Society

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$ k$-cobordism for links in $ S\sp 3$


Author: Tim D. Cochran
Journal: Trans. Amer. Math. Soc. 327 (1991), 641-654
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1991-1055569-2
MathSciNet review: 1055569
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Abstract: We give an explicit finite set of (based) links which generates, under connected sum, the $ k$-cobordism classes of links. We show that the union of these generating sets, $ 2 \leq k < \infty $, is not a generating set for $ \omega $-cobordism classes or even $ \infty $-cobordism classes.

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


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1055569-2
Keywords: Link, concordance, cobordism, $ \bar \mu $-invariants, Massey products
Article copyright: © Copyright 1991 American Mathematical Society

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