A Strong Haken Theorem

Suppose M = A U T B is a Heegaard split compact orientable 3 -manifold and S C M is a reducing sphere for M . Haken (1968) showed that there is then also a reducing sphere S * for the Heegaard splitting. Casson and Gordon (1987) extended the result to @ -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 .