simplicial complex

An abstract simplicial complex $K$ is a collection of nonemptyfinite sets with the property that for any element $\sigma \in K$, if$\tau \subset \sigma$ is a nonempty subset, then $\tau \in K$. Anelement of $K$ of cardinality $n+1$ is called an $n$-simplex. An element of an element of $K$ is called a vertex. In what follows, we may occasionally identify a vertex $v$ with its corresponding singleton set $\{v\}\in K$; the readerwill be alerted when this is the case.

The standard $n$-complex, denoted by ${\mathrm{\Delta }}_n$, is thesimplicial complex consisting of all nonempty subsets of$\{0,1,\mathrm{\dots },n\}$.

1 Geometry of a simplicial complex

Let $K$ be a simplicial complex, and let $V$ be the set of vertices of$K$. Although there is an established notion of infinite simplicialcomplexes, the geometrical treatment of simplicial complexes is much simpler in the finite case and so fornow we will assume that $V$ is a finite set of cardinality $k$.

We introduce the vector space ${ℝ}^V$ of formal$ℝ$–linear combinations of elements of $V$; i.e.,

$${ℝ}^V:=\{a_1{V}_1+a_2{V}_2+\mathrm{\cdots }+a_k{V}_k\mid a_i\in ℝ,V_i\in V\},$$

and the vector space operations are defined by formal addition andscalar multiplication. Note that we may regard each vertex in $V$ as aone-term formal sum, and thus as a point in ${ℝ}^V$.

The geometric realization of $K$, denoted $|K|$, is the subsetof ${ℝ}^V$ consisting of the union, over all $\sigma \in K$, ofthe convex hull of $\sigma \subset {ℝ}^V$. If we fix a bijection $\varphi :V\to \{1,\mathrm{\dots },k\}$, then the vector space ${ℝ}^V$ is isomorphic to the Euclidean vector space ${ℝ}^k$ via $\varphi$, and the set $|K|$inherits a metric from ${ℝ}^k$ making it into a metric spaceand topological space. The isometry class of $K$ is independent of the choice of the bijection $\varphi$.

Examples:

1. ${\mathrm{\Delta }}_2=\{\{0\},\{1\},\{2\},\{0,1\},\{0,2\},\{1,2\},\{0,1,2\}\}$ has $V=3$, so its realization $|{\mathrm{\Delta }}_2|$ is a subsetof ${ℝ}^3$, consisting of all points on the hyperplane $x+y+z=1$ that are inside or on the boundary of the first octant. Thesepoints form a triangle in ${ℝ}^3$ with one face, three edges,and three vertices (for example, the convex hull of $\{0,1\}\in {\mathrm{\Delta }}_2$ is the edge of this triangle that lies in the $xy$–plane).

2. Similarly, the realization of the standard $n$–simplex${\mathrm{\Delta }}_n$ is an $n$–dimensional tetrahedron contained inside${ℝ}^{n+1}$.

3. A triangle without interior (a “wire frame” triangle) can begeometrically realized by starting from the simplicial complex$\{\{0\},\{1\},\{2\},\{0,1\},\{0,2\},\{1,2\}\}$.

Notice that, under this procedure, an element of $K$ of cardinality 1is geometrically a vertex; an element of cardinality 2 is an edge;cardinality 3, a face; and, in general, an element of cardinality $n$is realized as an $n$-face inside ${ℝ}^V$.

In general, a triangulation of a topological space $X$ is asimplicial complex $K$ together with a homeomorphism from $|K|$ to$X$.

2 Homology of a simplicial complex

In this section we define the homology and cohomology groupsassociated to a simplicial complex $K$. We do so not because thehomology of a simplicial complex is so intrinsically interesting inand of itself, but because the resulting homology theory is identicalto the singular homology of the associated topological space $|K|$,and therefore provides an accessible way to calculate the latterhomology groups (and, by extension, the homology of any space $X$admitting a triangulation by $K$).

As before, let $K$ be a simplicial complex, and let $V$ be the set ofvertices in $K$. Let the chain group $C_n(K)$ be the subgroup of theexterior algebra $\mathrm{\Lambda }({ℝ}^V)$ generated by all elements of the form$V_0\wedge V_1\wedge \mathrm{\cdots }\wedge V_n$ such that $V_i\in V$ and$\{V_0,V_1,\mathrm{\dots },V_n\}\in K$. Note that we are ignoring here the$ℝ$–vector space structure of ${ℝ}^V$; the group $C_n(K)$ under thisdefinition is merely a free abelian group, generated by thealternating products of the above form and with the relations that areimplied by the properties of the wedge product.

Define the boundary map ${\partial }_n:C_n(K)⟶C_{n-1}(K)$ by the formula

$${\partial }_n(V_0\wedge V_1\wedge \mathrm{\cdots }\wedge V_n):=\sum _{j=0}^n{(-1)}^j(V_0\wedge \mathrm{\cdots }\wedge \widehat{V_j}\wedge \mathrm{\cdots }\wedge V_n),$$

where the hat notation means the term under the hat is left out of theproduct, and extending linearly to all of $C_n(K)$. Then one checkseasily that ${\partial }_{n-1}\circ {\partial }_n=0$, so the collectionof chain groups $C_n(K)$ and boundary maps ${\partial }_n$ forms a chaincomplex $𝒞(K)$. The simplicial homology and cohomologygroups of $K$ are defined to be that of $𝒞(K)$.

Theorem: The simplicial homology and cohomology groups of $K$,as defined above, are canonically isomorphic to the singular homologyand cohomology groups of the geometric realization $|K|$ of $K$.

The proof of this theorem is considerably more difficult than what wehave done to this point, requiring the techniques of barycentricsubdivision and simplicial approximation, and we refer the interestedreader to [1].