Delta Complexes

$\Delta$-Complexes and Chains

Standard $n$-simplex

The standard $n$-simplex, denoted ${\Delta }^n$, is a subspace of ${\mathbb{R}}^{n+1}$ defined as: ${\Delta }^n=\left\{(t_0,t_1,\dots ,t_n)\in {\mathbb{R}}^{n+1}\mid t_i\ge 0\text{ for all }i,\text{ and }{\sum }_{i=0}^n{t}_i=1\right\}.$

e.g. ${\Delta }^0$ is a point; ${\Delta }^1$ is a line segment; ${\Delta }^2$ is a triangle; ${\Delta }^3$ is a tetrahedron.

$\Delta$-Complex

A $\Delta$-complex is a topological space $X$ constructed by gluing together simplices (${\Delta }^k$ for various $k$) along their faces. The gluing maps must be linear and preserve the ordering of vertices.

Remark

One can think of it as a more structured and restrictive type of CW complex, where the cells are specifically simplices and the attaching maps have stricter rules.

e.g. We can build a torus using a $\Delta$-complex structure with the following components:

• One 0-simplex: a single vertex, $V$.
• Three 1-simplicies: edges $a,b,c$.
• Two 2-cells: simplicies $U,L$.

Imagine two triangles, $U$ and $L$. We glue their corresponding edges to form a rectangle, and then identify the opposite sides of this rectangle to form the torus:

Chain Group

The $n$-th chain group, $C_n^{\Delta }(X)$, is the free abelian group generated by the set of $n$-simplices in the $\Delta$-complex $X$. An element of $C_n^{\Delta }(X)$, called an $n$-chain, is a finite formal sum of $n$-simplices: $\sum n_i{\Delta }_i^n$, where $n_i\in \mathbb{Z}$ and ${\Delta }_i^n$ are $n$-simplices.

Boundary Operator

We define a homomorphism, the boundary operator, $d_n:C_n^{\Delta }(X)\to C_{n-1}^{\Delta }(X)$. If we denote an $n$-simplex by its ordered vertices $[v_0,v_1,\dots ,v_n]$, the boundary operator is defined by the alternating sum of its faces: $d_n([v_0,\dots ,v_n])={\sum }_{i=0}^n(-1{)}^i[v_0,\dots ,{\hat{v}}_i,\dots ,v_n]$ where $[v_0,\dots ,{\hat{v}}_i,\dots ,v_n]$ is the $(n-1)$-face opposite the vertex $v_i$. By convention, we set $d_0=0$. i.e., $\ker d_0=C_0^{\Delta }(X)$.

e.g.

• Boundary of an edge (1-simplex): $d_1([v_0,v_1])=[v_1]-[v_0]$. This is the “endpoint minus the start point”. For a loop, where $v_0=v_1$, the boundary is $0$.
• Boundary of a triangle (2-simplex): $d_2([v_0,v_1,v_2])=[v_1,v_2]-[v_0,v_2]+[v_0,v_1]$. This represents the sum of the oriented edges forming the boundary of the triangle.

Simplicial Homology Groups

The boundary operator has a crucial property that allows us to define homology.

Lemma

The boundary of a boundary is zero. That is, the composition of any two successive boundary maps is the zero map: $d_n\circ d_{n+1}=0$

Corollary

The image of $d_{n+1}$ is a subgroup of the kernel of $d_n$: $\mathrm{im}(d_{n+1})\le \ker (d_n)$.

This relationship lets us define the homology groups, which measure the “holes” in our space.

Simplicial Homology Group

The $n$-th simplicial homology group of $X$, denoted $H_n^{\Delta }(X)$, is the quotient group: $H_n^{\Delta }(X)=\frac{\text{ker}(d_n)}{\text{im}(d_{n+1})}$ The elements of $\text{ker}(d_n)$ are called $n$-cycles, and the elements of $\text{im}(d_{n+1})$ are called $n$-boundaries.

Remark

Homology captures the cycles that are not boundaries.

e.g. Recall our $\Delta$-complex structure for the torus $T^2$: The chain groups are: $C_2^{\Delta }(T)=\mathbb{Z}\{U,L\}$, $C_1^{\Delta }(T)=\mathbb{Z}\{a,b,c\}$, $C_0^{\Delta }(T)=\mathbb{Z}\{v\}$. The boundary maps are:

• $d_2:C_2(T)\to C_1(T)$, $U\mapsto a-c+b$, $L\mapsto b-c+a$.
• Since all edges $a,b,c$ are loops starting and ending at $v$, their boundaries are zero: $d_1(a)=v-v=0$, $d_1(b)=0$, $d_1(c)=0$.

$H_0(T)$:

• $\text{ker}(d_0)=C_0^{\Delta }(T)=\mathbb{Z}\{v\}$.
• $\text{im}(d_1)=\{0\}$.
• $H_0^{\Delta }(T)=\frac{\mathbb{Z}\{v\}}{\{0\}}\cong \mathbb{Z}$. This reflects the single connected component of the torus.

$H_1(T)$:

• $\text{ker}(d_1)=C_1(T)=\mathbb{Z}\{a,b,c\}$, since the boundary of every 1-chain is 0.
• $\text{im}(d_2)=\{k_1{d}_2(U)+k_2{d}_2(L)\}=\{(k_1+k_2)(a+b-c)\}$. This is the subgroup generated by $a+b-c$.
• $H_1^{\Delta }(T)=\frac{\mathbb{Z}\{a,b,c\}}{⟨a+b-c⟩}$. We can use the relation $c=a+b$ to eliminate $c$, leaving two independent generators, $a$ and $b$. Thus, $H_1^{\Delta }(T)\cong \mathbb{Z}\oplus \mathbb{Z}={\mathbb{Z}}^2$. These correspond to the two fundamental loops of the torus.

$H_2(T)$:

• $\text{ker}(d_2)$: An element $k_{1}U+k_{2}L$ is in the kernel if $d_2(k_{1}U+k_{2}L)=(k_1+k_2)(a+b-c)=0$. This implies $k_1=-k_2$. So, the kernel is the subgroup generated by the cycle $U-L$.
• $\text{im}(d_3)=0$, since there are no 3-cells in our complex.
• $H_2^{\Delta }(T)=\frac{\mathbb{Z}\{U-L\}}{\{0\}}\cong \mathbb{Z}$. This represents the 2D “void” enclosed by the torus surface.