Cobordism of linked discs.
(Cobordisme d’enlacements de disques.)

*(French)*Zbl 0666.57015The set of concordance classes of \(k\)-component spherical \(n\)-links (locally flat embeddings of \(kS^ n\) in \(S^{n+2})\) does not have a natural group structure except when \(k=1\). In this memoir this defect is ameliorated by considering first \((k,n)\)-disk links, which are proper embeddings of \(k\) copies of \(D^ n=[0,1]^ n\) in \(D^{n+2}\) which agree with a fixed trivial embedding on the boundary. Addition of such links may be defined by “stacking” with respect to the last co-ordinate. A further advantage of disk links is that they are better suited to study via homological surgery.

Chapter 0 reviews a theory of localization of CW-complexes due to Vogel, and the theory of homological surgery due to Cappell and Shaneson. (As Vogel’s work on localization has not been published elsewhere, more details are given in an appendix to the memoir).

In Chapter I \((k,n)\)-disk links and the group \(C_{n,k}\) of concordance classes of such links are defined, and it is shown that \(C_{n,k}\) is abelian if \(n>1\) (or if \(k=1)\). However, in Section 2 of this chapter it is shown hat \(C_{1,k}\) contains the group of pure \(k\)-string braids, and so is non-abelian if \(k>2\). This section also describes a group \(\mathcal H(G_ k)\) of automorphisms of the Vogel localization \(G_ k\) of the free group of rank \(k\), which is used in Chapter III. The final section gives some computations to test whether \(C_{1,2}\) is abelian, but does not settle this question.

The main result of the memoir is presented in Chapter II. There is a long exact sequence of abelian groups which traps the concordance groups \(C_{n,k}\) for \(n>2\) between homotopy groups and \(\Gamma\)-groups. Computing these other groups and the maps between them remains a formidable problem.

Chapter III begins with a discussion of boundary disk links; the group of boundary concordance classes of such links can be computed more explicitly, and is 0 if the (ambient) dimension is even. The remainder of the chapter considers spherical links. It is easy to see that every \(k\)-component spherical \(n\)-link may be obtained from a \((k,n)\)-disk link by taking the union along the boundary with a trivial \((k,n)\)-disk link. (In general this representation is far from unique). This leads to a surjection from \(C_{n,k}\) onto the set \(\tilde C_{n,k}\) of concordance classes of such spherical links. The main result of this section is the identification of \(\tilde C_{n,k}\) as the set of orbits of a natural action of the group \({\mathcal H}(G_ k)\) on \(C_{n,k}\) (for \(n>2)\).

Chapter 0 reviews a theory of localization of CW-complexes due to Vogel, and the theory of homological surgery due to Cappell and Shaneson. (As Vogel’s work on localization has not been published elsewhere, more details are given in an appendix to the memoir).

In Chapter I \((k,n)\)-disk links and the group \(C_{n,k}\) of concordance classes of such links are defined, and it is shown that \(C_{n,k}\) is abelian if \(n>1\) (or if \(k=1)\). However, in Section 2 of this chapter it is shown hat \(C_{1,k}\) contains the group of pure \(k\)-string braids, and so is non-abelian if \(k>2\). This section also describes a group \(\mathcal H(G_ k)\) of automorphisms of the Vogel localization \(G_ k\) of the free group of rank \(k\), which is used in Chapter III. The final section gives some computations to test whether \(C_{1,2}\) is abelian, but does not settle this question.

The main result of the memoir is presented in Chapter II. There is a long exact sequence of abelian groups which traps the concordance groups \(C_{n,k}\) for \(n>2\) between homotopy groups and \(\Gamma\)-groups. Computing these other groups and the maps between them remains a formidable problem.

Chapter III begins with a discussion of boundary disk links; the group of boundary concordance classes of such links can be computed more explicitly, and is 0 if the (ambient) dimension is even. The remainder of the chapter considers spherical links. It is easy to see that every \(k\)-component spherical \(n\)-link may be obtained from a \((k,n)\)-disk link by taking the union along the boundary with a trivial \((k,n)\)-disk link. (In general this representation is far from unique). This leads to a surjection from \(C_{n,k}\) onto the set \(\tilde C_{n,k}\) of concordance classes of such spherical links. The main result of this section is the identification of \(\tilde C_{n,k}\) as the set of orbits of a natural action of the group \({\mathcal H}(G_ k)\) on \(C_{n,k}\) (for \(n>2)\).

Reviewer: Jonathan A. Hillman (Sydney)

##### MSC:

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |

##### Keywords:

concordance classes of k-component spherical n-links; (k,n)-disk links; localization of CW-complexes; homological surgery; boundary disk links##### References:

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