Simplicial complex

Simplicial complexes are structures composed of simplices in a particular way. They provide a bridge between abstract and topological structures.

Definition

Abstract

An abstract simplicial complex is a set of finite sets π such that:

• ${\displaystyle x\subseteq y\land y\in {\mathcal {K}}\implies x\in {\mathcal {K}}}$.

That is it is a set which is closed under subsets.

Geometric

A geometric simplicial complex is a set of simplices π, such that:

• If ${\displaystyle x\in {\mathcal {K}}}$ then every element of x  is in π,
• The intersection of two simplices, x  and y , in π is an element of both x  and y .

Terminology

There are a number of terms used commonly with simplicial complexes which are unique or conflict with similar terms used for polytopes.

Cell
A rank k  simplex of a simplicial complex realized in k -dimensional space.
Closed star
The closed star denoted ${\displaystyle \operatorname {St} }$ is the closure of the star.
Closure
If π is a simplicial complex, then the closure of ${\displaystyle S\subset {\mathcal {K}}}$, denoted ${\displaystyle \operatorname {Cl} S}$, is the smallest subcomplex of π containing S .
Face
An element of a simplex.
Facet
A simplex in a complex that is not an element of another simplex in that complex.
Homogeneous
A k -complex where every simplex is an element of a k -simplex. Alternatively: A simplex where every facet has the same rank. Pure complexes may also be called pure.
Polyhedron
The space described by a simplicial complex.
Pure
Homogeneous
Star
If π is a simplicial complex, then the star of ${\displaystyle S\subset {\mathcal {K}}}$, denoted ${\displaystyle \operatorname {st} S}$, is the smallest set of all simplices in π that intersect with S .

Applications in polytopes

While simplicial complexes are not themselves polytopes, simplicial complexes give a method of deriving topological properties for polytopes which don't necessarily have an induced topology, including abstract, skew, and self-intersecting polytopes.

For a polytope π we construct an analogous abstract simplicial complex π by taking the elements of π as vertices in π. The simplicies in π are then sets of mutually incident elements. π is then a homogeneous k -complex where every flag of π corresponds to a k -simplex in π attached to other simplices along a k-1 -element iff those two flags are adjacent.

The simplicial complex can then be converted to an analogous cell complex to create a topology of π.