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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. 
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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 13:38, 28 February 2014

If you have the desire to process settings swiftly, loading files instantly, however, the body is logy plus torpid, what would we do? If you are a giant "switchboard" that is lack of efficient administration program and powerful housekeeper, what would we do? If you have send a exact commands to a notice, nevertheless the body could not do correctly, what would we do? Yes! We want a full-featured repair registry!

You have to understand certain easy plus inexpensive methods that may resolve the problem of the computer and speed it up. The earlier you fix it, the less damage the computer gets. I can tell regarding certain helpful techniques which can assist we to speed up you computer.

Registry cleaning is significant because the registry may get crowded plus messy when it's left unchecked. False entries send the running system shopping for files plus directories that have long ago been deleted. This takes time and utilizes valuable resources. So, a slowdown inevitably occurs. It is especially noticeable whenever we multitask.

Your computer was really rapidly when we initially purchased it. Because a registry was really clean plus without mistakes. After time, your computer starts to run slow plus freezes up today plus then. Because there are errors accumulating in it and certain information is rewritten or completely deleted by your incorrect uninstall of programs, wrong operations, malware or different things. That is the reason why the computer performance decreases slowly plus become especially unstable.

Google Chrome crashes on Windows 7 when the registry entries are improperly modified. Missing registry keys or registry keys with wrong values will cause runtime mistakes and thereby the issue happens. You are recommended to scan the entire system registry and review the outcome. Attempt the registry repair task utilizing third-party tuneup utilities software.

Although I constantly utilize the latest version of browser, often different extensions and plugins become the cause of errors with my browser plus the system. The same is the story with my browser that was crashing frequently possibly due to the Flash player error.

Why this is significant, is because most 'dumb' registry products actually delete these files without even understanding. They simply browse through the registry and try plus discover the many difficulties possible. They then delete any files they see fit, and because they are 'dumb', they don't really care. This means that when they delete some of these vital system files, they are really going to result a LOT more harm than advantageous.

By changing the means we utilize the internet you can have access more of the valuable bandwidth. This usually eventually provide we a faster surfing experience. Here is a link to 3 ways to customize the PC speed online.

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