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'''Pseudomanifold''' is a special type of [[topological space]].
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
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>]]
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
A pseudomanifold can be considered as a [[combinatorial]] realisation of the general idea of a [[manifold]] with [[Mathematical singularity|singularities]].  
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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>
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== Definition ==
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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>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


# {{nowrap|1=''X'' = {{!}}''K''{{!}}}} is the [[union (set theory)|union]] of all ''n''-[[simplex|simplices]].
<!--'''PNG''' (currently default in production)
# Every {{nowrap|1=(''n'' – 1)-simplex}} is a [[Euclidean_simplex#Faces|face]] of exactly two ''n''-simplices for ''n > 1''.
:<math forcemathmode="png">E=mc^2</math>
# 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 ===
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


*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>
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
*Condition 3 means that ''X'' is a '''strongly connected''' simplicial complex.<ref name="ANO"/>


== Examples ==
==Demos==
*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"/>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


* [[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"/>
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


== References ==
==Test pages ==


{{Reflist}}
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


[[Category:Topological spaces]]
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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