Simplicial Complex

For computation of fundamental groups of compact subsets of $\mathbb{E} ^n$ it is convenient to represent the subsets as simplicial complexes, made up from suitably joined copies of the standard n-simplices [2,4]. The standard n-simplex is

$$\Delta ^{n} \ =\ \{ (x_i)\in \mathbb{R} ^{n+1} \mid \sum x_i = 1,\ x_{i} \in [0,1]\ \forall\, i \}$$

and its vertices are the n+1 points

$$v_{0} = (1,0,\ldots ,0), \ v_{1}= (0,1,0,\ldots ,0),\ldots , v_{n} = (0,\ldots ,0,1)\, .$$

Conversely, given n+1 points $v_{0}, v_{1},\ldots ,v_{n}$ in ${\mathbb{R} }^{n+1}$ such that the n vectors $\{ v_{i} - v_{0} \mid i = 1,\ldots ,n \}$ are linearly independent, then they define an n-simplex. It is the subset:

$$(v_0, v_{1},\ldots ,v_{n}) = \{ x \in \mathbb{E} ^{n+1} \mid x = \sum t_{i} v_{i}, \ t_{i} \in [0,1], \ \sum t_{i} = 1 \}$$

which we say spans  the vertices $v_{0}, v_{1},\ldots ,v_{n}$ with barycentric coordinates  (t_i). A geometric (finite) simplicial complex K is a (finite) collection $\{ \sigma_{i} \in \mathbb{E} ^{m} \mid i=1,2,\ldots ,p \}$ of simplices, all in $\mathbb{E} ^{m}$ for some finite m, satisfying:

(i) $\sigma_{i} \cap \sigma_{j}$ is a face of $\sigma_{i}$ and of $\sigma_{j}$
(ii) every face of a simplex in K is itself a simplex in K. A simplicial complex K inherits the subspace topology and we denote this topological space by |K|. It is called a realization of K. Then a space X, homeomorphic to |K| is called a polyhedron and we say that K is a triangulation of X. Van Kampen's Theorem allows us to compute fundamental groups of spaces in terms of those of constituent simpler subspaces [2,4]:

Let M be a polyhedron with a triangulation as the union of two simplicial complexes $M\equiv J\cup K$ where $J,K,J\cap K$ are all path-connected with inclusions of the underlying spaces

$$\vert J\vert \stackrel{j}{\hookleftarrow} \vert J\cap K\vert \stackrel{i} {\hookrightarrow} \vert K\vert\, .$$

Then, for all vertices v in $J\cap K$,

$$\pi_1(\vert M\vert ,v) \cong \pi_1(\vert J\vert ,v)\ast\pi_1(\vert K\vert ,v)/{\sim}$$

where $\sim$ denotes the set of relations:

$$j_*(z) = k_*(z) \ \ \ \ \mbox{ for \ all}\ z\in \pi_1(\vert J\cap K\vert ,v)\,.$$

The simplicial complex structures actually generate a family of groups, called simplicial homology groups [2,4], one for each dimension present in the complex. These groups are easier to compute than the homotopy groups but they do in a sense approximate the latter.