Disk surgery and the primitive disk complexes of the 3-sphere

Abstract

It is known that the primitive disk complex for a genus-2 Heegaard splitting of the 3-sphere is closed under disk surgery operation. We show that, for a genus-g Heegaard splitting of the 3-sphere with g >= 3, the primitive disk complex for the splitting is not weakly closed under disk surgery operation. That is, there exist two primitive disks in one of the handlebodies of the splitting such that any disk surgery on one along the other one yields no primitive disks.