Elliptic surfaces without 1-handles

TL;DR

This paper proves that certain elliptic surfaces can be decomposed into handle structures without 1-handles, confirming a conjecture for specific cases and expanding understanding of their topology.

Contribution

It demonstrates that elliptic surfaces E(n)_{p,q} have handle decompositions without 1-handles for specific parameters, addressing a conjecture in 4-manifold topology.

Findings

Elliptic surfaces E(n)_{p,q} lack 1-handles for specified (p,q) values.

The result holds for all n ≥ 1 with (p,q) = (2,3), (2,5), (3,4), (4,5).

Confirms the Harer-Kas-Kirby conjecture in these cases.

Abstract

Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1)_{2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n)_{p,q} has a handle decomposition without 1-handles for $n\geq 1$ and (p,q)=(2,3),(2,5),(3,4),(4,5).