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A note on smoothly slice links in $S^2 \times S^2$

Marco Marengon, Clayton McDonald

TL;DR

The paper provides a smooth-category proof that there exist 2-component links not smoothly slice in $S^2\times S^2$, extending earlier work and highlighting potential for detecting exotic 4-manifolds. It combines genus, Arf, and Levine-Tristram obstruction techniques, including a detailed case analysis of homology classes and root-of-unity signatures, to rule out all possible sliceness scenarios. A concrete example is exhibited using a mirror of a knot, and the work indicates a pathway to constructing exotic $S^2\times S^2$ via surgery along such links with appropriate spin rational-bounding structures. The results emphasize the interplay between smooth topology, 4-manifold invariants, and link concordance in the search for exotic smooth structures in dimension four.

Abstract

We give an alternative proof of a result of Miyazaki and Yasuhara that there exists links that are not smoothly slice in $S^2 \times S^2$. We discuss potential applications to the detection of exotic $S^2 \times S^2$. This is a follow-up note to a similar paper for the $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ case.

A note on smoothly slice links in $S^2 \times S^2$

TL;DR

The paper provides a smooth-category proof that there exist 2-component links not smoothly slice in , extending earlier work and highlighting potential for detecting exotic 4-manifolds. It combines genus, Arf, and Levine-Tristram obstruction techniques, including a detailed case analysis of homology classes and root-of-unity signatures, to rule out all possible sliceness scenarios. A concrete example is exhibited using a mirror of a knot, and the work indicates a pathway to constructing exotic via surgery along such links with appropriate spin rational-bounding structures. The results emphasize the interplay between smooth topology, 4-manifold invariants, and link concordance in the search for exotic smooth structures in dimension four.

Abstract

We give an alternative proof of a result of Miyazaki and Yasuhara that there exists links that are not smoothly slice in . We discuss potential applications to the detection of exotic . This is a follow-up note to a similar paper for the case.
Paper Structure (15 sections, 13 theorems, 11 equations, 2 figures, 1 table)

This paper contains 15 sections, 13 theorems, 11 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Review of some obstructive methods
  4. Levine-Tristram signatures
  5. Arf invariant
  6. A 2-component link not smoothly slice in S2 x S2
  7. The structure of the link L
  8. The genus function on S2 x S2
  9. Symmetries
  10. Linking number and Arf invariant
  11. The genus function again
  12. Levine-Tristram signatures
  13. Proof of Theorem \ref{['thm:S2xS2']}
  14. The search for exotic S2 x S2's
  15. Assumptions for the link not smoothly slice in S2 x S2

Key Result

Theorem 1.1

The 2-component link in Figure fig:S2xS2link is not smoothly slice in $S^2 \times S^2$.

Figures (2)

  • Figure 1: A 2-component link which is not smoothly slice in $S^2 \times S^2$.
  • Figure 2: The structure of the link $L$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 2.1: V:G-signatureG:G-signature, see MMP for this statement
  • Theorem 2.2: L:signatures
  • Theorem 2.3: K:topologyY:CL, see MMP for this statement
  • Lemma 3.1: MM:CP2CP2bar
  • Theorem 3.2: R:minimalgenus
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 10 more