Theory

Delta complexes

The $\Delta$-complex structure is a generalization of simplicial complex structure. Instead of gluing simplices from the same ${\mathbb{R}}^{n+1}$ by causing them to intersect at faces, one glues simplices from disjoint copies of ${\mathbb{R}}^{n+1}$ using a formal quotient topology construction.

This technique allows many common spaces to be represented using fewer simplices. For example, a circle may be given as a $\Delta$-complex by identifying the two vertices of the standard $1$-simplex. The torus may be given using just two $2$-simplices by identifying each of the pairs of sides.

Why not simplicial?

These spaces cannot be formed as simplicial complexes by gluing the given simplices.

They are not literally simplicial complexes because the circle is round and the torus is not a subset of ${\mathbb{R}}^2$.

They are not homeomorphic to simplicial complexes (with the indicated decompositions) because the circle has a 1-loop while the torus has simplices which intersect in more than one face. (Both are impossible features for the homeomorphism type of simplicial complexes with the given decompositions.)

$\Delta$-Complex - Combinatorial

A combinatorial $\Delta$-complex is a sequence of simplex sets $\{S_n{\}}_{n\in \mathbb{N}}$ together with attaching maps $d_i^n:S_{n+1}\to S_n$, for $i=0,1,\dots ,n+1$, one for each facet of a simplex in $S_{n+1}$.

Attaching maps join the faces of each simplex to simplices of lower dimension. This works geometrically when the attaching maps preserve adjacency of faces of faces. Preserving adjacency is equivalent to this combinatorial condition:

$$d_i^n\circ d_j^{n+1}=d_{j-1}^n\circ d_i^{n+1}\text{when}i<j$$

The adjacency condition may be illustrated with a 2-simplex:

Faces are numbered by removing a vertex in the order of vertices. So $[\widehat{v_0},v_1,v_2]$ is edge 0 of the 2-simplex $[v_0,v_1,v_2]$, and $[v_0,\widehat{v_1},v_2]$ is edge 1, and so on. Thus $v_1$ is face 0 of the 1-simplex $[v_0,v_1]$, and $v_0$ is face 1, and so on for the other edges.

Now interpret the adjacency condition:
Take as input the 2-simplex.

• With $i=0,j=1$
  • The condition says “vertex 0 of edge 1 equals vertex $1-1$ of edge 0.”
  • These vertices are the point $v_2$.
• With $i=0,j=2$
  • The condition says “vertex 0 of edge 2 equals vertex $2-1$ of edge 0.”
  • These vertices are the point $v_1$.
• With $i=1,j=2$
  • The condition says “vertex 1 of edge 2 equals vertex $2-1$ of edge 1.”
  • These vertices are the point $v_0$.

$\Delta$-Complex - Topological

A $\Delta$-complex $\mathcal{K}$ is the topological space formed from a combinatorial $\Delta$-complex $\mathcal{S}=\{S_n{\}}_{n\in \mathbb{N}}$ as the disjoint union of simplices enumerated by elements of $\mathcal{S}$, glued together according to the combinatorial attaching maps with canonically defined linear functions interpolating between glued vertices.

$$\mathcal{K}=\left(\underset{n\in \mathbb{N}}{⨆}{S}_n\times {\sigma }_n\right)/\sim$$

where $(\sigma ,{\iota }^{i}x)\sim (d_i\sigma ,x)$, with ${\iota }^i:{\sigma }_{n-1}\hookrightarrow {\sigma }_n$ is the inclusion of the $i^{\text{th}}$ face of the standard simplex ${\sigma }_n$, and $x\in {\sigma }_{n-1}$ is any point in that face.

Simplicial vs. $\Delta$ complexes

How best to think about simplicial vs. $\Delta$- complexes?

• Simplicial complex:
  • Collection of geometric simplices in the same background space ${\mathbb{R}}^{n+1}$.
  • A sub-simplex brought to touch another sub-simplex only in one congruent face, where they are glued.
• $\Delta$-complex:
  • Collection of abstract geometric simplices in distinct background spaces (disjoint union).
  • Simplices in $S_{n+1}$ are attached to the lower-dimensional structure by gluing in the faces of new $(n+1)$-simplices, preserving only the adjacency relation.

Underlying space

Given a complex $\mathcal{K}$ (simplicial complex or $\Delta$-complex), the underlying topological space of $\mathcal{K}$ is denoted $|\mathcal{K}|$.

$\Delta$-homology

Homology may be defined for $\Delta$-complexes using the same technique as for simplicial complexes.

When a simplicial complex is also a $\Delta$-complex, their homology groups are canonically isomorphic.

Nerve complex

Nerve complex

Suppose $\mathcal{U}=\{U_i{\}}_{i\in I}$ is a collection of sets indexed by $I$. The nerve complex $\mathcal{N}(\mathcal{U})$ is the collection of finite subsets $J\subset I$ such that $\bigcap _{j\in J}{U}_j$ is nonempty.

That is, the nerve is the complex of indices of finite collections of mutually overlapping subsets from $\mathcal{U}$.

Notes:

• The definition does not assume that the $U_i$ are subsets of a topological space. If they are, we do not assume that they are open.
• The nerve complex is a combinatorial complex: the elements are sets of indices, not the subsets themselves.
• Suppose $J\in \mathcal{N}(\mathcal{U})$ is a subset in the nerve.
  • Then all subsets of $J$ are also in the nerve.
  • I.e. the nerve is closed under taking subsets.
• Therefore: the nerve is a combinatorial simplicial complex.

As a simplicial complex:

• vertices of the nerve correspond to individual sets $U_i\in \mathcal{U}$
• $k$-simplices in the nerve correspond to nonempty subsets having $k$ elements

---

Theorem: Geometric realization

Every combinatorial simplicial complex with $N$ vertices can be realized as a geometric simplicial complex in ${\mathbb{R}}^N$.

In fact, every combinatorial simplicial complex with maximum simplex dimension equal to $d$ can be realized as a geometric simplicial complex in ${\mathbb{R}}^{2d+1}$.

Čech complex

Čech complex

Suppose $(X,d)$ is a metric space. Suppose $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ is a finite set of points in $X$.

Fix $r>0$. Define $U_i$ as the open balls:

$$U_i={\mathbf{B}}_r(p_i),r=1,2,\dots ,n$$

The nerve of $\mathcal{U}=\{U_1,U_2,\dots ,U_n\}$ is called the Čech complex of $\mathcal{P}$.

Illustration - Čech complexes of four points

Let $\mathcal{P}$ be four points in ${\mathbb{R}}^2$ as in the figure.

The Čech complex created using $r=2$ is 1-dimensional – three of the points are linked.

The Čech complex created using $r=5$ is 2-dimensional – the three linked points now span a 2-simplex. The fourth point is also connected.