Boundary-incompressible surface
Introduction 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.
See also * Incompressible surface
References * W. Jaco, ''Lectures on Three-Manifold Topology'', volume 43 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, R.I., 1980. * T. Kobayashi, ''A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth'', Osaka J. Math. '''29''' (1992), no. 4, 653–674..