Abstract Simplicial Complexes

Subjects: Topology
Links: Simplicial Complexes

Simplicial complexes were invented in the hope that they would enable topological questions about manifolds to be reduced to combinatorial questions simplicial complexes.

Def: We define an abstract simplicial complex to be a collection $\mathcal{K}$ of finite sets, subject to only one condition, to be hereditary.

If $\mathcal{K}$ is an abstract simplicial complex, the finite sets that make up $\mathcal{K}$ are called abstract simplices. Given an abstract simplex $s\in \mathcal{K}$, any element of $s$ is called a vertex of $s$, and any subset of $s$ is called a face of $s$. We say $\mathcal{K}$ is a finite complex if $\mathcal{K}$ itself it finite, and locally finite complex if every vertex belongs to only finitely many abstract simplices. The dimension of an abstract simplex $s\in \mathcal{K}$ is one less than the number of elements of $s$. If the dimensions of the abstract simplices of $\mathcal{K}$ are bounded above, them we say $\mathcal{K}$ is finite dimensional, and its dimension is the smallest upper bound of the dimensions of its simplices.

Now suppose that $\mathcal{K}$ and $\mathcal{L}$ are abstract complexes. Define their vertex sets by $${\cal K}0 := \bigcup{s\in \cal K}, \quad {\cal L}0 := \bigcup{s\in \cal L} s.$$
A map $f:\mathcal{K}\to \mathcal{L}$ is called an abstract simplicial map if it is of the form $f[\{v_0,\dots ,v_k\}]=\{f_0(v_0),\dots ,f_0(v_k)\}$ for some map $f_0:{\mathcal{K}}_{\mathcal{0}}\to {\mathcal{L}}_{\mathcal{0}}$, called the vertex map of $f$, that satisfies $f[s]\in \mathcal{L}$ for every $s\in \mathcal{K}$. An abstract simplicial map $f$ is called an isomorphism if both $f_0$ and $f$ are bijections. In this case $f^{-1}$ is also a simplicial map.

Given a Euclidean simplicial complex $K$, let $\mathcal{K}$ denote the collection of those finite sets $\{v_0,\dots ,v_k\}$ that consist of the vertices of some simplex $K$. It is immediate that $\mathcal{K}$ is an abstract simplicial complex called the vertex schema of $K$.

Obs: Two Euclidean complexes are simplicially isomorphic iff their vertex schema are isomorphic.

Def: If $K$ is a Euclidean simplicial complex vertex schema is isomorphic to $\mathcal{K}$, we say that $K$ is the geometric realisation of $\mathcal{K}$.

Prop: Every finite abstract simplicial complex has a geometric realisation.

Th: An abstract simplicial complex is the vertex schema of a Euclidean simplicial complex iff it is finite-dimensional, locally finite, and countable.

Def: Two simplicial complexes are said to be combinatorially equivalent if they have a common subdivision.

It was conjectured that if two simplicial complexes have homeomorphic polyhedra, they are combinatorially equivalent; this conjecture became known as the Hauptvermutung of combinatorial topology. We know that it is true for all complexes of dimension $2$ and for triangulated compact manifolds of dimension $3$, buy false in all higher dimensions, even for compact manifolds.

We can define certain how get subcomplexes. Let $\mathcal{K}$ be a simplicial complex.

• Let ${\mathcal{K}}^{\prime }\subseteq \mathcal{K}$, its closure is $\text{cl}({\mathcal{K}}^{\prime }):=\{\tau \in \mathcal{K}\mid \tau \subseteq \sigma \in {\mathcal{K}}^{\prime }\}$.
• The star of a simplex $\tau \in \mathcal{K}$ is $\text{st}(\tau ):=\sigma \in \mathcal{K}\mid \tau \subseteq \sigma \}$.
• The link of a simplex $\tau \in \mathcal{K}$ is $$\text{lk}(\tau) := {\sigma \in \text{cl}(\text{st}(\tau)) \mid \tau \cap \sigma = \varnothing}.$$