A Strong Haken's Theorem

Abstract

Suppose T is a Heegaard splitting surface for a compact orientable 3-manifold M, and 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 1-surgeries. Here we show that in fact one may take S* = S.

Figures (21)

• Figure 1: 2-handles and dual spine in a compression body.
• Figure 2: $\alpha'$ avoids $\Delta \cap A$
• Figure 3: Arcs $\alpha$ and $\beta$ both properly isotopic to $\gamma$
• Figure 4: Preamble to Lemma \ref{['lemma:vertdodge']}
• Figure 5: Concluding the proof of Lemma \ref{['lemma:vertdodge']}

Theorems & Definitions (71)

• Theorem 1.1: Haken, Casson-Gordon
• Definition 1.2
• Theorem 1.3
• Corollary 1.4: Strong Haken
• Proposition 2.1
• proof
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof