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In mathematics, an incompressible surface is a kind of two-dimensional surface inside of a 3-manifold.
To be precise, suppose that S is a compact surface properly embedded in a closed 3-manifold M. Suppose that D is a disk, also embedded in M, with
- <math> D \cap S = \partial D.<math>
Suppose finally that the curve <math>\partial D<math> in S does not bound a disk inside of S. Then D is called a compressing disk for S and we also call S a compressible surface in M. If no such disk exists then we call S incompressible.
An important consequence of incompressibility follows from the loop theorem. Let <math>\iota: S \rightarrow M<math> be an embedding of a two-sided properly embedded compact surface. Then the induced map on fundamental groups <math>\iota_\star: \pi_1(S) \rightarrow \pi_1(M)<math> is injective if and only if the surface is incompressible.
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