Generalizing Simplicial Complexes

Motivation and Introduction

Simplices are the building blocks of TDA that combine to give a simplicial complex approximation of the underlying space $X$. One of the most commonly observed ideas in mathematics is to generalize mathematical objects of interest. In other words, we wish to “peel off” a subset of the object’s properties to understand the newly obtained “peeled-off” object’s structure. For example, topological spaces are a generalization of metric spaces.

In the current context, this idea leads to the question - Are simplicial complexes the special case of something more general? The answer is yes! The most generalized building blocks in TDA are called cells which are glued together to form topological spaces called CW-complexes, which will be the object of interest in this blog. This blog will also cover the concept of a Polytopal Complex, a generalization of the simplicial complex, and an interesting special case of the CW-complex. The primary reference to understanding CW-complexes is Prof. Jonny Evans’ video lecture 4.01¹ on CW-complexes.

Note: The blog is mainly for an intuitive understanding with (minimal) mathematical rigor just enough to get an idea of the mathematical structure of CW-complexes. For a more rigorous mathematical approach to cell complexes and cellular homology refer to the textbook Organized Collapse: An Introduction to Discrete Morese Theory by Dmitry N. Kozlov².

Quotient Topology preliminaries

Understanding quotient topology is essential to understanding the construction of CW complexes. Lecture video 3.01 by Dr. Jonny Evans² provides the necessary mathematical intuition behind quotient topology. Chapter 5 in the topology textbook by Dr. Marco Manetti³ provides a rigorous and detailed introduction to quotient topology. Note that the topology textbook by Munkres⁴ is a popular introductory topology textbook in general.

Intuitively, quotient topology is a mathematically rigorous way of “gluing/pasting”. Consider the closed interval $[0,1]$, for instance. “Gluing” the ends $0$ and $1$ gives a circle. One will need the concept of quotient topology to prove this statement mathematically.

We define an equivalence relation to glue points together.

Generalizing the Construction of Simplicial Complex

Consider the construction of a simple abstract simplicial complex $K$ whose geometric realization is a tetrahedron in $\mathbb{R}^3$. A pictorial representation of its construction is given below.

\[K = \left \lbrace \ \emptyset, \langle a \rangle, \langle b \rangle, \langle c \rangle, \langle d \rangle, \langle a,b \rangle, \langle a,c \rangle, \langle a,d \rangle, \langle b,c \rangle, \langle b,d \rangle, \langle c,d \rangle, \langle a,b,c \rangle, \langle a,b,d \rangle, \langle b,c,d \rangle, \langle a,c,d \rangle, \langle a,b,c,d \rangle \ \right \rbrace\]

• Step 0: Start with $K = { \emptyset }$ i.e. the empty simplicial complex.
• Step 1: Add 0-simplices or vertices $\langle a \rangle, \langle b \rangle, \langle c \rangle, \langle d \rangle$ to the complex.
• Step 2: Add $1$-simplices or edges corresponding every pair of vertices. Consider the vertices $\langle a \rangle, \langle b \rangle, \langle c \rangle$. The edges $\langle a,b \rangle$ and $\langle b,c \rangle$ are distinct but the edges are glued at their common vertex or boundary point $\langle b \rangle$ — similarly the other vertices and their corresponding edges.
• Step 3: Add faces or 2-simplices $\langle p, q, r \rangle$ corresponding to every $3$-tuple of edges. Two faces, \langle a, b, c \rangle and \langle a, b, d \rangle for example, are glued together at one of their boundaries which is the edge $\langle a, b \rangle$. The same follows for other pairs of faces with a common edge.
• Step 4: Add the 3-simplex, which is the complete tetrahedron itself.

Polytopal complexes, Simplicial complexes and CW complexes