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Non-smoothable surfaces in the 4-sphere

Anthony Conway, Daniel Galvin

Abstract

We construct examples of non-smoothable surfaces in the $4$-sphere, thereby answering Question 4.32 on the K3 problem list. These surfaces are non-orientable and have knot group of order $2$, thus simultaneously answering Question 4.29(a) on the K3 problem list.

Non-smoothable surfaces in the 4-sphere

Abstract

We construct examples of non-smoothable surfaces in the -sphere, thereby answering Question 4.32 on the K3 problem list. These surfaces are non-orientable and have knot group of order , thus simultaneously answering Question 4.29(a) on the K3 problem list.
Paper Structure (15 sections, 17 theorems, 42 equations)

This paper contains 15 sections, 17 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Further context and comparisons
  4. Realisation and stable realisation of non-degenerate forms
  5. The exterior non-degenerate form
  6. Pullbacks of hermitian forms over $\mathbb{Z}[\mathbb{Z}_2]$.
  7. Pullbacks
  8. Pullbacks and equivariant intersection forms
  9. Constructing non-smoothable surfaces
  10. Building the stable isometry $\alpha_-$ of the minus forms
  11. Building the stable isometry $\alpha_+$ of the plus forms
  12. Stably realising $\lambda_b$
  13. Realising the form by a surface
  14. Neighbouring lattices
  15. Root-counting

Key Result

Theorem 1.1

There are infinitely many non-smoothable knotted $\mathbb{Z}_2$-surfaces in $S^4$, pairwise distinct up to isotopy.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • proof : Proof of Theorem \ref{['thm:Main']}
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Example 3.2
  • ...and 39 more