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 Σ in a homology ball X such that ∂(X, Σ) = (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. * ,(1) 3 b * ,(2) 3 b * ,(3) 3 b * ,(4) 3 b * ,(4) 4Proof. This follows from the earlier discussion.