Emission theory: Difference between revisions
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In [[mathematics]], an '''incompressible surface''', in intuitive terms, is a [[surface]], embedded in a [[3-manifold]], which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold. | |||
For a precise definition, suppose that ''S'' is a [[compact surface]] [[Embedding#Differential_topology|properly embedded]] in a 3-manifold ''M''. Suppose that ''D'' is a [[disk (mathematics)|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 and ''S'' is not the [[2-sphere]], then we call ''S'' '''incompressible''' (or '''geometrically incompressible'''). | |||
Note that we must exclude the 2-sphere to get any interesting consequences for the 3-manifold. Every 3-manifold has many embedded 2-spheres, and a 2-sphere embedded in a 3-manifold never has a compressing disc. | |||
Sometimes one defines an '''incompressible sphere''' to be a 2-sphere in a 3-manifold that does not bound a 3-ball. Thus, such a sphere either does not separate the 3-manifold or gives a nontrivial [[connected sum]] decomposition. Since this notion of incompressibility for a sphere is quite different from the above definition for surfaces, often an incompressible sphere is instead referred to as an '''essential sphere''' or '''reducing sphere'''. | |||
There is also an algebraic version of incompressibility: Suppose <math>\iota: S \rightarrow M</math> is a proper embedding of a compact surface. Then ''S'' is '''<math>\pi_1</math>-injective''' (or '''algebraically incompressible''') if the induced map on [[fundamental group]]s | |||
:<math>\iota_\star: \pi_1(S) \rightarrow \pi_1(M)</math> | |||
is [[injective]]. | |||
In general, every <math>\pi_1</math>-injective surface is incompressible, but the reverse implication is not always true. For instance, the [[Lens space]] <math> L(4,1) </math> contains an incompressible Klein bottle that is not <math>\pi_1</math>-injective. However, if <math> S </math> is a [[two-sided]] properly embedded, compact surface (not a 2-sphere), the [[loop theorem]] implies <math> S </math> is incompressible if and only if it is <math>\pi_1</math>-injective. | |||
==See also== | |||
* [[Compression (3-manifold)]] | |||
* [[Haken manifold]] | |||
* [[Virtually Haken conjecture]] | |||
* [[Thurston norm]] | |||
* [[Boundary-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. | |||
{{DEFAULTSORT:Incompressible Surface}} | |||
[[Category:3-manifolds]] | |||
Revision as of 21:16, 6 January 2014
In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold.
For a precise definition, suppose that S is a compact surface properly embedded in a 3-manifold M. Suppose that D is a disk, also embedded in M, with
Suppose finally that the curve 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 and S is not the 2-sphere, then we call S incompressible (or geometrically incompressible).
Note that we must exclude the 2-sphere to get any interesting consequences for the 3-manifold. Every 3-manifold has many embedded 2-spheres, and a 2-sphere embedded in a 3-manifold never has a compressing disc.
Sometimes one defines an incompressible sphere to be a 2-sphere in a 3-manifold that does not bound a 3-ball. Thus, such a sphere either does not separate the 3-manifold or gives a nontrivial connected sum decomposition. Since this notion of incompressibility for a sphere is quite different from the above definition for surfaces, often an incompressible sphere is instead referred to as an essential sphere or reducing sphere.
There is also an algebraic version of incompressibility: Suppose is a proper embedding of a compact surface. Then S is -injective (or algebraically incompressible) if the induced map on fundamental groups
is injective.
In general, every -injective surface is incompressible, but the reverse implication is not always true. For instance, the Lens space contains an incompressible Klein bottle that is not -injective. However, if is a two-sided properly embedded, compact surface (not a 2-sphere), the loop theorem implies is incompressible if and only if it is -injective.
See also
- Compression (3-manifold)
- Haken manifold
- Virtually Haken conjecture
- Thurston norm
- Boundary-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.