Uniqueness in Haken’s Theorem

Abstract

Following Haken and Casson-Gordon, it was shown in [Sc] that given a reducing sphere or boundary-reducing disk S in a Heegaard split manifold M in which every sphere separates, the Heegaard surface T can be isotoped so that it intersects S in a single circle. Here we show that when this is achieved by two different positionings of T, one can be moved to the other by a sequence of 1) isotopies of T rel S 2) pushing a stabilizing pair of T through S and 3) eyegelass twists of T. The last move is inspired by one of Powell's proposed generators for the Goeritz group.