PL-genus of surfaces in homology balls

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.

19 pages, 3 figures