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'''Pseudomanifold''' is a special type of [[topological space]].  
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It looks like a manifold at most of the points, but may contain singularities.
For example, the cone of solutions of <math>z^2=x^2+y^2</math> forms a pseudomanifold.
 
[[File:Pinched torus.jpg|thumb|<center>A pinched torus</center>]]
A pseudomanifold can be considered as a [[combinatorial]] realisation of the general idea of a [[manifold]] with [[Mathematical singularity|singularities]].  
The concepts of [[orientability]], orientation and [[degree of a mapping]] make sense for pseudomanifolds and moreover, within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts.<ref>{{Citation|first=H.|last=Steifert|first2=W.|last2=Threlfall|title=Textbook of Topology|publisher=Academic Press Inc.|year=1980|ISBN=0-12-634850-2}}</ref><ref>{{Citation|first=H.|last=Spanier|title=Algebraic Topology|publisher=McGraw-Hill Education|year=1966|ISBN=0-07-059883-5}}</ref>
 
== Definition ==
 
A topological space ''X'' endowed with a [[Triangulation (topology)|triangulation]] ''K'' is an ''n''-dimensional pseudomanifold if the following conditions hold:<ref name="BRAS">{{cite journal |last1=Brasselet|first1=J. P.|year=1996 |title=Intersection of Algebraic Cycles |journal= Journal of Mathematical Sciences|publisher=Springer New York|volume= 82|issue= 5|pages=3625 − 3632|url=http://www.springerlink.com/content/ju28j2wqm174hx10}}</ref>
 
# (''pure'') {{nowrap|1=''X'' = {{!}}''K''{{!}}}} is the [[union (set theory)|union]] of all ''n''-[[simplex|simplices]].
# Every {{nowrap|1=(''n'' – 1)-simplex}} is a [[Euclidean_simplex#Faces|face]] of exactly two ''n''-simplices for ''n > 1''.
# For every pair of ''n''-simplices σ and σ' in ''K'', there is a [[sequence]] of ''n''-simplices {{nowrap|1=σ = σ<sub>0</sub>, σ<sub>1</sub>, &hellip;, σ<sub>''k''</sub> = σ'}} such that the [[intersection (mathematics)|intersection]] {{nowrap|1=σ<sub>''i''</sub> ∩ σ<sub>''i''+1</sub>}} is an {{nowrap|1=(''n'' − 1)-simplex}} for all ''i''.
 
=== Implications of the definition ===
 
*Condition 2 means that ''X'' is a '''non-branching''' [[simplicial complex]].<ref name="ANO">{{citeweb|url=http://eom.springer.de/p/p075720.htm|author=D. V. Anosov|title=Pseudo-manifold|accessdate=August 6, 2010}}</ref>
*Condition 3 means that ''X'' is a '''strongly connected''' simplicial complex.<ref name="ANO"/>
 
==Related definitions==
 
*A pseudomanifold is called ''normal'' if link of each simplex with codimension <math>\ge 2</math> is a pseudomanifold.
 
== Examples ==
*A [[Pinched Torus|pinched torus]] (see figure) is an example of an [[orientable surface|orientable]], [[compact surface|compact]] 2-dimensional pseudomanifold.<ref name="BRAS"/>
 
* Complex [[algebraic varieties]] (even with singularities) are examples of pseudomanifolds.<ref name="ANO"/>
 
* [[Thom space]]s of [[vector bundle]]s over triangulable [[compact manifold]]s are examples of pseudomanifolds.<ref name="ANO"/>
 
* Triangulable, [[compact space|compact]], [[connected space|connected]], [[homology manifold]]s over '''Z''' are examples of pseudomanifolds.<ref name="ANO"/>
 
== References ==
 
{{Reflist}}
 
[[Category:Topological spaces]]
[[Category:Manifolds]]

Latest revision as of 23:12, 21 December 2014

Dental veneers are exceedingly thin shells or laminates that could differ in composition. If you loved this article and you would like to receive more info relating to Wood veneers tampa i implore you to visit our website. Often times they're made from your tooth-colored ceramic, composite bonding material or porcelain. The material is cemented to the surface of your tooth to improve its appearance.

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You may have heard of yet another popular option called porcelain veneers. When an individual veneers suited for your teeth, the porcelain is manufactured to give the leading of your teeth a whole new surface. Anyone have experienced someone with that gorgeous celeb smile, put on weight a pretty good chance that others have had porcelain veneers applied at some time in their life.

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