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Classification of Torus Fibrations Over $S^2$ Up to Fibre Sum Stabilisation

Yibo Zhang

Abstract

We study torus fibrations over the 2-sphere and Hurwitz equivalence of their monodromies. We show that, if two torus fibrations over $S^2$ have the same type of singularities, then their global monodromies are Hurwitz equivalent after performing direct sums with certain torus Lefschetz fibrations. The additional torus Lefschetz fibration is universal when the type of singularities is "simple".

Classification of Torus Fibrations Over $S^2$ Up to Fibre Sum Stabilisation

Abstract

We study torus fibrations over the 2-sphere and Hurwitz equivalence of their monodromies. We show that, if two torus fibrations over have the same type of singularities, then their global monodromies are Hurwitz equivalent after performing direct sums with certain torus Lefschetz fibrations. The additional torus Lefschetz fibration is universal when the type of singularities is "simple".
Paper Structure (21 sections, 58 theorems, 157 equations, 2 tables)

This paper contains 21 sections, 58 theorems, 157 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Connected sums and Hurwitz equivalence
  3. Elementary transformations
  4. Contraction and restoration on tuple
  5. Direct sums of fibrations and their global monodromies
  6. Hurwitz equivalence of global monodromies
  7. Swappability and local models
  8. Swappability of the normal form
  9. Fibre-preserving homeomorphisms: from local to global
  10. Singular fibrations and singularities
  11. Local Milnor fibre of genus zero
  12. Local Milnor fibre of genus one
  13. Theorem of R. Livné, complement and extension
  14. Tuples of short elements
  15. Conjugates of short elements and tuples
  16. ...and 6 more sections

Key Result

Theorem A

Let $\mathcal{O}$ be a multi-set of conjugacy classes of $\mathop{\mathrm{SL}}\nolimits(2,\mathbb{Z})$. There exists a torus Lefschetz fibration $f_O^L$ over $S^2$ depending only on the non-simple conjugacy classes occurring in $\mathcal{O}$ that has the following property: for $i=1,2$, Then $(g_1^{(1)},\ldots,g_n^{(1)})$ and $(g_1^{(2)},\ldots,g_n^{(2)})$ are Hurwitz equivalent i.e. one can tran

Theorems & Definitions (135)

  • Definition
  • Definition
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Corollary A
  • Corollary B
  • Definition 2.1.1
  • Lemma 2.1.2
  • ...and 125 more