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In low-dimensional topology, a boundary-in … In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot be made simpler by a certain type of operation known as boundary compression. Suppose M is a 3-manifold with boundary. Suppose also that S is a compact surface with boundary that is properly embedded in M,meaning that the boundary of S is a subset of the boundary of M and the interior points of S are a subset of the interior points of M.A boundary-compressing disk for S in M is defined to be a disk D in M such that and are arcs in , with , , and is an essential arc in S ( does not cobound a disk in S with another arc in ). The surface S is said to be boundary-compressible if either S is a disk that cobounds a ball with a disk in or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible. Alternatively, one can relax this definition by dropping the requirement that the surface be properly embedded. Suppose now that S is a compact surface (with boundary) embedded in the boundary of a 3-manifold M. Suppose further that D is a properly embedded disk in M such that D intersects S in an essential arc (one that does not cobound a disk in S with another arc in ). Then D is called a boundary-compressing disk for S in M. As above, S is said to be boundary-compressible if either S is a disk in or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible. For instance, if K is a trefoil knot embedded in the boundary of a solid torus V and S is the closure of a small annular neighborhood of K in , then S is not properly embedded in V since the interior of S is not contained in the interior of V. However, S is embedded in and there does not exist a boundary-compressing disk for S in V, so S is boundary-incompressible by the second definition.y-incompressible by the second definition.
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In low-dimensional topology, a boundary-in … In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot be made simpler by a certain type of operation known as boundary compression. The surface S is said to be boundary-compressible if either S is a disk that cobounds a ball with a disk in or there exists a boundary-compressing disk for S in M. Otherwise, S is boundary-incompressible.. Otherwise, S is boundary-incompressible.
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Boundary-incompressible surface
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