Homology classes of negative square and embedded surfaces in 4-manifolds

TL;DR

This paper establishes bounds on the self-intersection numbers of embedded surfaces in simply-connected 4-manifolds, depending on their homology class properties, with implications for the existence of certain embedded spheres.

Contribution

It provides new bounds on negative self-intersection for embedded surfaces with fixed genus in 4-manifolds, especially for divisible or characteristic classes.

Findings

Bounded N for fixed genus g in certain homology classes

Lower bounds on self-intersection for embedded spheres

Addresses a problem from the Kirby list

Abstract

Let X be a simply-connected closed oriented 4-manifold and A an embedded surface of genus g and negative self-intersection -N. We show that for fixed genus g there is an upper bound on N if the homology class of A is divisible or characteristic. In particular, for genus zero, there is a lower bound on the self-intersection of embedded spheres in these kinds of homology classes. This question is related to a problem from the Kirby list.