simplicial complex

An abstract simplicial complexMathworldPlanetmath K is a collectionMathworldPlanetmath of nonempty finite setsMathworldPlanetmath with the property that for any element σK, if τσ is a nonempty subset, then τK. An element of K of cardinality n+1 is called an n-simplex. An element of an element of K is called a vertex. In what follows, we may occasionally identify a vertex v with its corresponding singleton set {v}K; the reader will be alerted when this is the case.

The standard n-complex, denoted by Δn, is the simplicial complex consisting of all nonempty subsets of {0,1,,n}.

1 Geometry of a simplicial complex

Let K be a simplicial complex, and let V be the set of vertices of K. Although there is an established notion of infiniteMathworldPlanetmath simplicial complexes, the geometrical treatment of simplicial complexes is much simpler in the finite case and so for now we will assume that V is a finite set of cardinality k.

We introduce the vector spaceMathworldPlanetmath V of formal linear combinationsMathworldPlanetmath of elements of V; i.e.,


and the vector space operationsMathworldPlanetmath are defined by formal additionPlanetmathPlanetmath and scalar multiplication. Note that we may regard each vertex in V as a one-term formal sum, and thus as a point in V.

The geometric realization of K, denoted |K|, is the subset of V consisting of the union, over all σK, of the convex hull of σV. If we fix a bijectionMathworldPlanetmath ϕ:V{1,,k}, then the vector space V is isomorphicPlanetmathPlanetmathPlanetmath to the Euclidean vector space k via ϕ, and the set |K| inherits a metric from k making it into a metric space and topological spaceMathworldPlanetmath. The isometryMathworldPlanetmath class of K is independent of the choice of the bijection ϕ.


  1. 1.

    Δ2={{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}} has V=3, so its realization |Δ2| is a subset of 3, consisting of all points on the hyperplaneMathworldPlanetmathPlanetmath x+y+z=1 that are inside or on the boundary of the first octant. These points form a triangleMathworldPlanetmath in 3 with one face, three edges, and three vertices (for example, the convex hull of {0,1}Δ2 is the edge of this triangle that lies in the xy–plane).

  2. 2.

    Similarly, the realization of the standard n–simplex Δn is an n–dimensional tetrahedronMathworldPlanetmathPlanetmath contained inside n+1.

  3. 3.

    A triangle without interior (a “wire frame” triangle) can be geometrically realized by starting from the simplicial complex {{0},{1},{2},{0,1},{0,2},{1,2}}.

Notice that, under this procedure, an element of K of cardinality 1 is geometrically a vertex; an element of cardinality 2 is an edge; cardinality 3, a face; and, in general, an element of cardinality n is realized as an n-face inside V.

In general, a triangulation of a topological space X is a simplicial complex K together with a homeomorphismMathworldPlanetmath from |K| to X.

2 Homology of a simplicial complex

In this sectionMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath we define the homologyMathworldPlanetmathPlanetmathPlanetmath and cohomology groupsPlanetmathPlanetmath associated to a simplicial complex K. We do so not because the homology of a simplicial complex is so intrinsically interesting in and of itself, but because the resulting homology theory is identical to the singular homology of the associated topological space |K|, and therefore provides an accessiblePlanetmathPlanetmath way to calculate the latter homology groups (and, by extensionPlanetmathPlanetmathPlanetmath, the homology of any space X admitting a triangulation by K).

As before, let K be a simplicial complex, and let V be the set of vertices in K. Let the chain group Cn(K) be the subgroupMathworldPlanetmathPlanetmath of the exterior algebra Λ(V) generated by all elements of the form V0V1Vn such that ViV and {V0,V1,,Vn}K. Note that we are ignoring here the –vector space structureMathworldPlanetmath of V; the group Cn(K) under this definition is merely a free abelian group, generated by the alternating productsPlanetmathPlanetmathPlanetmath of the above form and with the relationsMathworldPlanetmathPlanetmathPlanetmath that are implied by the properties of the wedge productMathworldPlanetmath.

Define the boundary map n:Cn(K)Cn-1(K) by the formulaMathworldPlanetmathPlanetmath


where the hat notation means the term under the hat is left out of the product, and extending linearly to all of Cn(K). Then one checks easily that n-1n=0, so the collection of chain groups Cn(K) and boundary maps n forms a chain complex 𝒞(K). The simplicial homology and cohomology groups of K are defined to be that of 𝒞(K).

TheoremMathworldPlanetmath: The simplicial homology and cohomology groups of K, as defined above, are canonically isomorphic to the singular homology and cohomology groups of the geometric realization |K| of K.

The proof of this theorem is considerably more difficult than what we have done to this point, requiring the techniques of barycentric subdivision and simplicial approximation, and we refer the interested reader to [1].


Title simplicial complex
Canonical name SimplicialComplex
Date of creation 2013-03-22 12:34:46
Last modified on 2013-03-22 12:34:46
Owner djao (24)
Last modified by djao (24)
Numerical id 11
Author djao (24)
Entry type Definition
Classification msc 55U10
Classification msc 54E99
Related topic HomologyTopologicalSpace
Related topic CWComplex
Defines simplicial homology
Defines simplicial cohomology
Defines triangulation
Defines abstract simplicial complex
Defines abstract n-simplex