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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 $M_1$ and $M_2$. 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 $S\cap M_i$ is incompressible for each $i=1,2$.
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참고문헌 (7)
참고문헌 신청
D. Bachman / 2010 / Topological index theory for surfaces in 3-manifolds / Geom. Topol 14(1):585~609
W. Haken / 1968 / Studies in Modern Topology / Math. Assoc. Amer, Prentice-Hall :39~98
J. Hempel / 2001 / 3-manifolds as viewed from the curve complex / Topology 40(3):631~657
K. Ichihara / Decomposing Heegaard splittings along sepa-rating incompressible surfaces in 3-manifolds / Topology Appl
W. Jaco / 1980 / Lectures on Three-Manifold Topology / American Mathematical Society
W. Haken / 1968 / Studies in Modern Topology / Math. Assoc. Amer, Prentice-Hall :39~98
J. Hempel / 2001 / 3-manifolds as viewed from the curve complex / Topology 40(3):631~657
K. Ichihara / Decomposing Heegaard splittings along sepa-rating incompressible surfaces in 3-manifolds / Topology Appl
W. Jaco / 1980 / Lectures on Three-Manifold Topology / American Mathematical Society
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