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 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
does not cobound a disk in S with another arc 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
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
and there does not exist a boundary-compressing disk for S in V, so S is boundary-incompressible by the second definition.