Editing Simplicial complex (section)

== Algebraic topology ==
{{Main|Simplicial homology}}
In [[algebraic topology]], simplicial complexes are often useful for concrete calculations. For the definition of [[homology group]]s of a simplicial complex, one can read the corresponding [[chain complex]] directly, provided that consistent orientations are made of all simplices. The requirements of [[homotopy theory]] lead to the use of more general spaces, the [[CW complex]]es. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at [[Polytope]] of simplicial complexes as subspaces of [[Euclidean space]] made up of subsets, each of which is a [[simplex]]. That somewhat more concrete concept is there attributed to [[Pavel Sergeevich Alexandrov|Alexandrov]]. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a [[Compact space|compact]] [[topological space]] which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a [[polyhedron]] (see {{harvnb|Spanier|1966}}, {{harvnb|Maunder|1996}}, {{harvnb|Hilton|Wylie|1967}}).