Abstract

We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links.

Highlights

  • The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice [7, p. 243]

  • A good boundary link is an m-component link L that admits a homomorphism φ : π1(S3 L) → F sending the meridians to the m generators of the free group F of rank m, such that the kernel of φ is perfect, or equivalently such that H1(S3 L; ZF) = 0

  • We say that a link L is freely slice if L bounds slicing discs in D4 whose complement has free fundamental group generated by the meridians of L

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Summary

Introduction

The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice [7, p. 243]. Theorem A Suppose that L is a good boundary link that has a Seifert surface admitting a homotopically trivial+ good basis. Observe that if {ai , bi } is a homotopically trivial+ good basis each of the sets of curves {ai } and {bi } is a derivative of L, that is a collection of mutually disjoint curves on a Seifert surface for L whose linking and self-linking numbers all vanish. All the entries of (1.1) indicated by a ∗ are 0 for a homotopically trivial+ good basis Boundary links admitting this kind of Seifert matrix were termed generalised doubles by Freedman and Krushkal in [5, Definition 5.9]. The fact that Whitehead doubles of homotopically trivial+ links are slice is a special case of Theorem A.

Seifert matrices of boundary links
Seifert matrices and S-equivalence
S-equivalence and good boundary links
Freely slicing a boundary link by finding a slice disc exterior
Construction of the 4-manifold W
Comparison with previously known results
Examples that are not Whitehead doubles
Questions
Full Text
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