Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds

TL;DR

The paper discusses how recent methods for constructing different smooth structures on 4-manifolds can produce infinite families of nullhomologous surfaces, which are topologically isotopic but smoothly distinct, within various 4-manifolds.

Contribution

It introduces a procedure to generate infinite sets of pairwise smoothly non-isotopic nullhomologous 2-tori and spheres in 4-manifolds, expanding understanding of smooth structures.

Findings

Infinite sets of smoothly non-isotopic nullhomologous surfaces constructed

Surfaces are topologically isotopic and sometimes bound a handlebody

Method applies to various 4-manifolds

Abstract

We point out that recent constructions of inequivalent smooth structures yield a manufacturing procedure of infinite sets of pairwise smoothly non-isotopic nullhomologous 2-tori and spheres inside a myriad of 4-manifolds. The corresponding infinite set consists of topologically isotopic surfaces that topologically bound a handlebody in several instances.