A strong Haken's theorem

Abstract: Suppose that \(T\) is a Heegaard splitting surface for a compact orientable three-manifold \(M\); suppose that \(S\) is a reducing sphere for \(M\). In 1968 Haken showed that there is then also a reducing sphere \(S^*\) for the Heegaard splitting. That is, \(S^*\) is a reducing sphere for \(M\) and the surfaces \(T\) and \(S^*\) intersect in a single circle. In 1987 Casson and Gordon extended the result to boundary-reducing disks in \(M\) and noted that in both cases \(S^*\) is obtained from \(S\) by a sequence of operations called one-surgeries. Here we show that in fact one may take \(S^* = S\), at least in the case where \(M\) contains no \(S^1 \times S^2\) summands.

geometric topology

Audience: researchers in the topic

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Geometry and topology online

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