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  "body": "\n## Summary\n\nAn incompressible surface is a [[surface|VGJJVBZ1MQ]] inside a [[3-manifold|R20TB2KEQQ]] that cannot be simplified by cutting it along a disk in the manifold.\nFor two-sided surfaces other than spheres and disks, this means that the inclusion of the surface induces an injective map on fundamental groups.\nIncompressible surfaces are the main tool to cut $3$-manifolds into simpler pieces, as in the theory of Haken manifolds and the JSJ decomposition.\n\n## Explanations\n\n### L1 human\n\nImagine a surface sitting inside a $3$-dimensional space.\nSuppose some loop on the surface bounds a disk in the surrounding space, but not on the surface.\nThen one can cut the surface along that loop and glue in two copies of the disk, making the surface simpler.\nA surface is incompressible when no such move is possible.\nThe boundary torus of a solid doughnut is compressible: the loop around the tube bounds a disk inside the doughnut.\nA torus $T^2 \\times \\set{1/2}$ inside the thickened torus $T^2 \\times [0, 1]$ is incompressible.\n",
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