Editing Simplicial complex (section)

== Triangulation == 
{{Main|Triangulation (topology)}}

A triangulation of a [[topological space]] <math>X</math> is a [[homeomorphism]] <math>t: |\mathcal{T}|\rightarrow X</math> where <math>\mathcal{T}</math> is a simplicial complex.

Topological spaces do not necessarily admit a triangulation and if they do, it is never unique. [[Topological manifold]]s of dimension <math>d \leq 3</math> are always triangulable,<ref>{{citation |surname1=Edwin Moise |title=Geometric Topology in Dimensions 2 and 3 |publisher=Springer Verlag |publication-place=New York |date=1977 }}</ref><ref>{{cite web |title=Über den Begriff der Riemannschen Fläche |periodical= |publisher= |url=https://www.maths.ed.ac.uk/~v1ranick/papers/rado.pdf |first=Tibor |last=Rado |language=German }}</ref><ref>{{citation |surname1=John M. Lee |title=Introduction to Topological manifolds |publisher=Springer Verlag |publication-place=New York/Berlin/Heidelberg |at=p.&nbsp;92 |isbn=0-387-98759-2 |date=2000 }}</ref> but not necessarily for <math>d > 3</math>.<ref>{{citation |author1=R. C. Kirby |author2=L. C. Siebenmann |periodical=Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88) |title=Annex B. On The Triangulation of Manifolds and the Hauptvermutung |publisher=Princeton University Press |at=pp.&nbsp;299–306 |date=1977-12-31 }}</ref><ref>{{cite book |last1=Akbulut |first1=Selman |last2=McCarthy |first2=John D. |title=Casson's Invariant for Oriented Homology Three-Spheres {{!}} Princeton University Press |date=19 April 2016 |chapter=Chapter IV: Casson's Invariant for Oriented Homology 3-spheres |publisher=Princeton University Press |isbn=9780691636085 |language=en }}</ref>

[[Differentiable manifold]]s of any dimension <math>d\ge 1</math> admit triangulations.<ref>{{citation |surname1=J. H. C. Whitehead |periodical=Annals of Mathematics |title=On C1-Complexes |volume=41 |issue=4 |at=pp.&nbsp;809–824 |issn=0003-486X |date=1940 |doi=10.2307/1968861 |jstor=1968861 }}</ref>