The fundamental building blocks of TDA are simplices that can be combined 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 structure of the newly obtained “peeled-off object”. 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 a special case of the CW-complex. The primary reference to understanding CW-complexes is Prof. Jonny Evans’ video lecture 4.01[^1] 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 Organized Collapse: An Introduction to Discrete Morese Theory by Dmitry N. Kozlov.
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.