Editing Incompressible surface (section)

== Compression ==

[[File:Compressing a surface along a disk.svg|right|360px|thumb|Compressing a surface {{math|''S''}} along a disk {{math|''D''}} results in a surface {{math|''S'''}}, which is obtained by removing the annulus boundary of {{math|''N''(''D'')}} from {{math|''S''}} and adding in the two disk boundaries of {{math|''N''(''D'').}}]]

Given a compressible surface {{math|''S''}} with a compressing disk {{math|''D''}} that we may assume lies in the [[Interior (topology)|interior]] of {{math|''M''}} and intersects {{math|''S''}} transversely, one may perform embedded 1-[[Surgery theory|surgery]] on {{math|''S''}} to get a surface that is obtained by '''compressing''' {{math|''S''}} '''along''' {{math|''D''}}.  There is a [[tubular neighborhood]] of {{math|''D''}} whose closure is an embedding of ''D'' &times; [-1,1] with ''D'' &times; 0 being identified with ''D'' and with
:<math>(D\times [-1,1])\cap S=\partial D\times [-1,1].</math>
Then
:<math>(S-\partial D\times(-1,1))\cup (D\times \{-1,1\})</math>
is a new properly embedded surface obtained by compressing {{math|''S''}} along {{math|''D''}}.

A non-negative complexity measure on compact surfaces without 2-sphere components is {{math|''b''<sub>0</sub>(''S'') &minus; ''&chi;''(''S'')}}, where {{math|''b''<sub>0</sub>(''S'')}} is the zeroth [[Betti number]] (the number of connected components) and {{math|''&chi;''(''S'')}} is the [[Euler characteristic]] of {{math|''S''}}.  When compressing a compressible surface along a nontrivial compressing disk, the Euler characteristic increases by two, while {{math|''b''<sub>0</sub>}} might remain the same or increase by 1.  Thus, every properly embedded compact surface without 2-sphere components is related to an incompressible surface through a sequence of compressions.

Sometimes we drop the condition that {{math|''S''}} be compressible.  If {{math|''D''}} were to bound a disk inside {{math|''S''}} (which is always the case if {{math|''S''}} is incompressible, for example), then compressing {{math|''S''}} along {{math|''D''}} would result in a disjoint union of a sphere and a surface homeomorphic to {{math|''S''}}.  The resulting surface with the sphere deleted might or might not be [[Homotopy#isotopy|isotopic]] to {{math|''S''}}, and it will be if {{math|''S''}} is incompressible and {{math|''M''}} is irreducible.