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On elliptic surfaces which have no 1-handles

Daisuke Kusuda

Abstract

Gompf conjectured that the elliptic surface $E(n)_{p,q}$ has no handle decomposition without 1- and 3-handles. We prove that each of the elliptic surfaces $E(n)_{5,6}$, $E(n)_{6,7}$, $E(n)_{7,8}$ and $E(n)_{8,9}$ has a handle decomposition without 1-handles for $n\geq4$, $n\geq 5$, $n\geq 9$ and $n\geq 24$, respectively.

On elliptic surfaces which have no 1-handles

Abstract

Gompf conjectured that the elliptic surface has no handle decomposition without 1- and 3-handles. We prove that each of the elliptic surfaces , , and has a handle decomposition without 1-handles for , , and , respectively.

Paper Structure

This paper contains 3 sections, 6 theorems, 5 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Logarithmic transformation
  3. Proof of theorem 1.2

Key Result

Theorem 1.2

Each of the elliptic surfaces $E(n)_{5,6}$, $E(n)_{6,7}$, $E(n)_{7,8}$ and $E(n)_{8,9}$ has a handle decomposition without 1-handles for $n\geq4$, $n\geq 5$, $n\geq 9$ and $n\geq 24$, respectively.

Figures (26)

  • Figure 1: Cusp neighborhood
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 21 more figures

Theorems & Definitions (11)

  • Theorem 1.2
  • Lemma 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 1 more