On elliptic surfaces which have no 1-handles

TL;DR

This paper proves that certain elliptic surfaces, specifically $E(n)_{p,q}$ for particular values of p, q, and n, admit handle decompositions without 1-handles, supporting Gompf's conjecture in these cases.

Contribution

The authors establish the existence of handle decompositions without 1-handles for specific families of elliptic surfaces, advancing understanding of their topological structures.

Findings

Elliptic surfaces $E(n)_{p,q}$ have no 1-handles for specified p, q, and n.

Proved for $E(n)_{5,6}$ with $n extgreater 4$.

Proved for $E(n)_{6,7}$ with $n extgreater 5$.

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.