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Abstract Simplicial Complex

Definitions

An abstract simplicial complex is a pair $K=(V,\Sigma)$ where $V$ is called the set of vertices and $\Sigma \subseteq \mathcal{P}(V)$, called the set of simplices, is such that

• $\emptyset \notin \Sigma$
• $\forall v \in V: {v} \in \Sigma$
• if $\sigma \in \Sigma$ then also $\tau \in \Sigma$ for all $\tau \subset \sigma: \tau \ne \emptyset$.

A simplex with n+1 vertices is called an n-simplex.

Properties

An edge path is a sequence of vertices $v_i \in V: 0 \le i \le n$ such that $\{v_i, v_i+1\} \in \Sigma$ for $0 \le i < n$. In the topological realisation, this is related to the connectedness property.

Types

A simplicial circle is a simplicial complex of 0- and 1-simplices, with vertices $\{v_1, ..., v_n\}$ where $n \ge 3$ and 1-simplices $\{v_i, v_{i+1}\}: 1 \le i < n$ and $\{v_n, v_1\}$.

If K is finite and the link of every vertex is a simplicial circle, then the simplicial complex is called a closed combinatorial surface. Such a s.c. contains simplices of dimension at most 2, and every 1-simplex is contained in exactly two 2-simplices. Its topological realisation is a closed surface.

Examples

A subcomplex $K'=(V',\Sigma')$ of $K=(V, \Sigma)$ is a simplicial complex where $V' \subset V$ and $\Sigma' \subset \Sigma$

The standard n-simplex is represented by a simplex with n+1 vertices. Its faces, vertices, etc. are represented by the simplicial complex $(V, \mathcal{P}(V))$ where $|V|=n+1$.

all these pages adapted with probably insufficient credit from my university's lecture notes