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Boundary Dehn twists on Milnor fibers and Family Bauer--Furuta invariants

Jin Miyazawa

Abstract

We proved that the boundary Dehn twist on the Milnor fiber $M_c(2, q, r)$ is an exotic diffeomorphism relative to the boundary if $q, r$ are odd, coprime integers bigger than $3$ and $(q-1)(r-1)/4$ is an odd number. The proof is given by comparing the family relative Bauer--Furuta invariants of the mapping torus.

Boundary Dehn twists on Milnor fibers and Family Bauer--Furuta invariants

Abstract

We proved that the boundary Dehn twist on the Milnor fiber is an exotic diffeomorphism relative to the boundary if are odd, coprime integers bigger than and is an odd number. The proof is given by comparing the family relative Bauer--Furuta invariants of the mapping torus.

Paper Structure

This paper contains 10 sections, 7 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Review of the family Bauer--Furuta invariant
  3. Family on $S^1$
  4. Definitions of our invariants and calculations
  5. Definition of the $\mathbb{Z}/2\mathbb{Z}$-invariant: case \ref{['closure']}
  6. Definition of the $\mathbb{Z}/2\mathbb{Z}$-invariant: case \ref{['symplectic']}
  7. The choice of a stable framing of $H^+(\mathbb{X})$ and the family invariant
  8. Exotic diffeomorphisms
  9. Proof of the main theorem
  10. Examples

Key Result

Theorem 1.1

Let $M_c(p, q, r)$ be the Milnor fiber that is given by for some small $\epsilon \in \mathbb{C} \setminus 0$. If $p=2$ and $q, r$ are odd, coprime, and is an odd number, then the boundary Dehn twist on $M_c(2, q, r)$ is an exotic diffeomorphism relative to the boundary.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • ...and 8 more