Bicompressible surfaces and incompressible surfaces

摘要

We give new evidence that ``complicated'' Heegaard surfaces behave like incompressible surfaces. More precisely, suppose that a closed connected orientable 3-manifold M contains a closed connected incompressible surface F which separates M into two (connected) components M1 and M2. Let S be a Heegaard surface of M. Our result is that if the Hempel distance of S is at least four, then S is isotoped so that S∩Mi is incompressible for each i=1,2.