Abstract Simplicial Complex - Geometric Realization

We can associate to an abstract simplicial complex K a topological space |K|, called its geometric realization, which is a simplicial complex. The construction goes as follows.

First, define |K| as a subset of ^S consisting of functions t:S → satisfying the two conditions:

Now think of ^S as the direct limit of ^A where A ranges over finite subsets of S, and give ^S the induced topology. Now give |K| the subspace topology.

Alternatively, let denote the category whose objects are the faces of K and whose morphisms are inclusions. Next choose a total order on the vertex set of K and define a functor F from to the category of topological spaces as follows. For any face X ∈ K of dimension n, let F(X) = Δn be the standard n-simplex. The order on the vertex set then specifies a unique bijection between the elements of X and vertices of Δn, ordered in the usual way e₀ < e₁ < ... < e_n. If Y ⊂ X is a face of dimension m < n, then this bijection specifies a unique m-dimensional face of Δn. Define F(Y) → F(X) to be the unique affine linear embedding of Δm as that distinguished face of Δn, such that the map on vertices is order preserving.

We can then define the geometric realization |K| as the colimit of the functor F. More specifically |K| is the quotient space of the disjoint union

by the equivalence relation which identifies a point y ∈ F(Y) with its image under the map F(Y) → F(X), for every inclusion Y ⊂ X.

If K is finite, then we can describe |K| more simply. Choose an embedding of the vertex set of K as an affinely independent subset of some Euclidean space RN of sufficiently high dimension N. Then any face X ∈ K can be identified with the geometric simplex in RN spanned by the corresponding embedded vertices. Take |K| to be the union of all such simplices.

If K is the standard combinatorial n-simplex, then clearly |K| can be naturally identified with Δn.