PL-genus of surfaces in homology balls

TL;DR

This paper demonstrates that the minimal PL genus of surfaces in homology balls bounding a homology sphere-knot pair can be arbitrarily large, using Heegaard Floer homology techniques.

Contribution

It establishes the unboundedness of the minimal PL genus for surfaces in homology balls bounding homology sphere-knot pairs, a new insight in 3-manifold topology.

Findings

Minimal PL genus can be arbitrarily large.

Uses Heegaard Floer homology to prove unboundedness.

Connects surface genus in homology balls to knot cobordisms.

Abstract

We consider manifold-knot pairs $(Y,K)$ where $Y$ is a homology sphere that bounds a homology ball. We show that the minimum genus of a PL surface $\Sigma$ in a homology ball $X$ such that $\partial (X, \Sigma) = (Y, K)$ can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from $(Y, K)$ to any knot in $S^3$ can be arbitrarily large. The proof relies on Heegaard Floer homology.