Simplicial complex

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Definition

A simplicial complex, [ilmath]K[/ilmath], in [ilmath]\mathbb{R}^N[/ilmath] is a collection of simplices, [ilmath]K[/ilmath], such that:

1. [ilmath]\forall \sigma\in K\forall\tau\in[/ilmath][ilmath]\text{Faces}(\sigma)[/ilmath][ilmath][\tau\in K][/ilmath]
2. [ilmath]\forall \sigma,\tau\in K[\sigma\cap\tau\neq\emptyset\implies(\sigma\cap\tau\in\text{Faces}(\sigma)\wedge\sigma\cap\tau\in\text{Faces}(\tau))][/ilmath]
   • TODO: "The intersection of any two simplices is a face in each of them" is what he says, [ilmath]\emptyset[/ilmath] being a face would tidy this up slightly but I still think it is not a face!

Underlying set & topology

We use [ilmath]\vert K\vert [/ilmath] to denote the "underlying set" of [ilmath]K[/ilmath]:

• [ilmath]\vert K\vert:\eq\bigcup_{\sigma\in K}\sigma[/ilmath] - as expected

To make [ilmath]\vert K\vert[/ilmath] into a topological space we require a topology, say [ilmath]\mathcal{J} [/ilmath] (so [ilmath](\vert K\vert,\mathcal{ J })[/ilmath] is a topological space)

• [ilmath]\mathcal{J}:\eq\left\{U\in\mathcal{P}(\vert K\vert)\ \vert\ \forall\sigma\in K[\sigma\cap U\text{ open in }\sigma]\right\} [/ilmath] - recall [ilmath]\mathcal{J} [/ilmath] is the set of open sets of the topological space.
  • Equivalently: [ilmath]\mathcal{J}:\eq\left\{U\in\mathcal{P}(\vert K\vert)\ \vert\ \forall\sigma\in K\exists V\in\mathcal{K}[\sigma\cap U\eq \sigma\cap V]\right\} [/ilmath] where [ilmath]\mathcal{K} [/ilmath] is the topology of [ilmath]\mathbb{R}^N[/ilmath] - the usual topology from the Euclidean metric
    • Recall a simplex has the subspace topology for its topology.
    • TODO: Confirm a simplex has the subspace topology!
• We can also work with closed sets:
  • [ilmath]A\in\mathcal{P}(\vert K\vert)[/ilmath] is closed if and only if [ilmath]\forall\sigma\in K[\sigma\cap A\text{ is closed in }\sigma][/ilmath]

Terminology

• The underlying set, [ilmath]\vert K\vert[/ilmath] is sometimes called the polytope of [ilmath]K[/ilmath]
  • A space that is the polytope of a simplicial complex may be called a polyhedron - but some topologists reserve this for the polytope of a finite simplicial complex

TODO: We are undecided on this

Comments

• The topology of [ilmath]\vert K\vert[/ilmath] may be finer than the topology [ilmath]\vert K\vert[/ilmath] would inherit as a subspace of [ilmath]\mathbb{R}^N[/ilmath]. We form the following claim:

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The claim Munkres makes is:

• Suppose [ilmath]A[/ilmath] is closed in [ilmath]\vert K\vert[/ilmath] considered as a topological subspace of [ilmath]\mathbb{R}^N[/ilmath] then [ilmath]A[/ilmath] is closed in in [ilmath]\vert K\vert[/ilmath] with its topology as defined above.
  • i.e. [ilmath]\text{closed in subspace}\implies\text{closed in space} [/ilmath]

What does this mean for open sets?

Equivalent definitions

• A collection of simplices, [ilmath]K[/ilmath], is a simplicial complex iff [ilmath]\forall\sigma\in K\forall \tau\in\text{Faces}(\sigma)[\tau\in K][/ilmath] and [ilmath]\forall\sigma,\tau\in K[\sigma\neq\tau\implies \text{Int}(\sigma)\cap\text{Int}(\tau)\eq\emptyset][/ilmath]

Properties

• A map from a simplicial complex to a space is continuous if and only if the map restricted to each simplex in the complex is continuous also
  • Barycentric coordinate with respect to a point of a simplicial complex
• A simplicial complex is a Hausdorff space
• If a simplicial complex is finite then it is compact
• If a subset of a simplicial complex is compact then that subset is a subset of a finite subcomplex of the complex

TODO: There's more and clean up!

See also

• Simplicial subcomplex (Subcomplex (simplex) redirects there) - usual sub construction as encountered everywhere
• Simplicial p-skeleton
• Vertex (simplicial complex)
• Abstracit simplicial complex
• Simplex