Mathematics Colloquia and Seminars

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Suppose $M$ is a Heegaard split compact orientable $3$-manifold and $S \subset M$ is a reducing sphere for $M$. Haken showed that there is then also a reducing sphere $S^*$ for the Heegaard splitting; that is, $S^*$ intersects the Heegaard surface in a single essential circle. Casson and Gordon extended the result to $\partial$-reducing disks in $M$ and noted that in both cases $S^*$ is obtained from $S$ by a sequence of operations called $1$-surgeries. Here we show that in fact one may take $S^* = S$