3 Simplicial complexes
“Do you see the leaves falling from the trees, the sun rising and setting?”
— Alfred de Musset, in The Confession of a Child of the Century
3.1 The infinite through a window
A circle, a torus, and the real line contain infinitely many points. A computer has a finite amount of memory, and so do I. We need a finite description that still remembers something about the shape.
As Section 2.4 showed, a finite metric point cloud has a discrete topology. Just storing a list of points gives us no loops. This does not make every finite topological space uninteresting—non-discrete finite topologies exist—but it explains why a list of measurements needs another construction.
Mathematics often handles the infinite with finite descriptions. Induction uses a starting case and a rule for the next case. A finite-dimensional vector space uses a finite basis to describe infinitely many vectors. Simplicial complexes offer a similar bargain: finitely many building blocks can describe a space with infinitely many points.
3.2 Graphs
Euler studied the bridges of Königsberg in 1736 because he wanted to visit the city but was too lazy1 to walk around like a normal person. Could he cross each bridge exactly once?
Forget the geography for a moment. Represent each land region by a vertex and each bridge by an edge:
An edge records a pairwise relation. The bridge example is a multigraph, because two land regions can have more than one bridge between them. Most graphs in this book are simpler:
Definition 3.1 A simple undirected graph is a pair \(G=(V,E)\), with vertex set \(V\) and edges that are two-element subsets \(\{v,w\}\subseteq V\). There are no repeated edges or edges from a vertex to itself.
A directed graph instead uses ordered pairs \((v,w)\), often written \(v\to w\). Unless stated otherwise, our graphs are undirected.
At every intermediate stop on a bridge-crossing walk, an arrival must pair with a departure. Thus at most two vertices—the start and the finish—can have odd degree. All four land regions in the Königsberg graph have odd degree, so the proposed walk is impossible. A finite description has answered the question without making Euler wear out his shoes.
3.2.1 The essence of a circle
A geometric circle is the set of points a fixed distance from a center. Topologically, it is also what we get by joining the ends of an interval.
The drawing has three vertices and three edges. The geometric realization of this graph includes every point along the edges, not just the three vertices; it is homeomorphic to a circle. There is no filled triangular face. That missing face will become very important.
3.3 Simplices and their faces
Why stop with vertices and edges? An edge uses two vertices; three can specify a filled triangle, four a filled tetrahedron, and so on. Congratulations, you have reinvented simplicial complexes!
Definition 3.2 A finite abstract simplicial complex \(K\) on a vertex set \(V\) is a collection of subsets of \(V\) such that every subset of a member of \(K\) also belongs to \(K\).
A nonempty member \(\sigma\in K\) is a simplex. If \(\sigma\) has \(k+1\) vertices, its dimension is \(k\). Its subsets are its faces. We include the empty face by convention, but do not count it when computing ordinary homology.
Thus a \(0\)-simplex is a vertex, a \(1\)-simplex an edge, a \(2\)-simplex a filled triangle, and a \(3\)-simplex a filled tetrahedron. Including \(\{a,b,c\}\) requires including its three edges and three vertices. The converse fails: having all three edges does not force an abstract complex to contain the triangle.
The intersection of two simplices is automatically a face of both. This follows from closure under subsets; we do not need a second axiom for the abstract definition.
3.4 Abstract objects, geometric spaces
The definition is a list of subsets; it says nothing about coordinates. To turn this list into a space, realize each simplex as a solid geometric simplex and glue matching faces.
For \(n\) vertices, there is a convenient construction in \(\mathbb R^n\). Assign vertex \(i\) the standard basis vector \(e_i\). For every \(\sigma\in K\), take the convex hull of its assigned vertices. Their union is the geometric realization, written \(|K|\):
\[ |K|=\left\{\sum_{i=1}^n t_i e_i\ \middle|\ t_i\geq0,\ \sum_i t_i=1, \ \{i:t_i>0\}\in K\right\}. \]
A simplex has only finitely many vertices, but its geometric realization has infinitely many points. There is our window onto the infinite.
We can often draw the same realization in a smaller ambient space. However, the coordinates of the original data need not realize the abstract complex faithfully. Four coplanar square corners cannot be the vertices of a nondegenerate tetrahedron! When the Rips construction below includes a \(3\)-simplex, it means an abstract tetrahedron; it does not suddenly lift one of our measured points off the table.
We use the combinatorial view for computation and the geometric view for intuition. Results about the topology of a complex refer to its realization (Hatcher 2002).
3.5 From data to simplicial complexes
Let \(X=\{x_1,\ldots,x_n\}\) be a finite set in a metric space. Distances tell us which vertices are near one another. A scale parameter turns that information into simplices.
3.5.1 Vietoris–Rips complex
Definition 3.3 For \(\epsilon\geq0\), the Vietoris–Rips complex \(\mathrm{VR}(X,\epsilon)\) consists of the subsets \(\sigma\subseteq X\) such that
\[ d(x,x')\leq\epsilon\qquad\text{for every }x,x'\in\sigma. \]
We use the diameter convention: \(\epsilon\) is the largest allowed pairwise distance, not a ball radius. Some references use a different factor of two, so always check the convention before comparing scales.
First build a graph linking points at distance at most \(\epsilon\). Then fill every clique: three mutually linked vertices give a triangle, four give a tetrahedron, and so on. This automatically includes all faces. Notice the difference from the graph in Figure 3.1: in a Rips complex, a complete three-vertex graph comes with its filling.
3.5.2 Čech complex
Fix an ambient metric space containing \(X\), and place a closed ball of radius \(\epsilon/2\) at each data point. The Čech complex contains \(\sigma\) when those balls have a common intersection:
\[ \bigcap_{x\in\sigma}\overline B(x,\epsilon/2)\neq\emptyset. \]
The ambient space matters: an intersection in \(\mathbb R^2\) is different from an intersection restricted to the finite set \(X\).
In Euclidean space, these balls and their nonempty intersections are convex, hence contractible. The nerve theorem for a finite family of convex sets implies that the Čech realization has the homotopy type of their union (Edelsbrunner and Harer 2010). “Same homotopy type” is weaker than “homeomorphic,” but it is enough to preserve homology.
The corresponding general theorem uses a good cover: nonempty finite intersections must be contractible, together with appropriate cover hypotheses, such as a finite open cover. Balls in an arbitrary metric space do not automatically satisfy this condition. The Čech definition still makes sense there; the Euclidean conclusion does not follow merely because we named the sets balls.
Rips needs only pairwise distances; Čech requires testing common intersections. This helps explain why Rips is convenient for general data, while Čech and its Euclidean relatives offer a closer connection to unions of balls. We revisit the computational choices in Computing persistent homology.
3.6 Our square at three scales
Label the square corners in cyclic order as in Section 2.4. Their Euclidean distance matrix is
\[ D=\begin{array}{c|cccc} &a&b&c&d\\\hline a&0&1&\sqrt2&1\\ b&1&0&1&\sqrt2\\ c&\sqrt2&1&0&1\\ d&1&\sqrt2&1&0 \end{array}. \]
The sides have length \(1\); the diagonals have length \(\sqrt2\). We can therefore list every change in the Rips complex without guessing:
| Scale | Simplices present | Shape of the realization |
|---|---|---|
| \(0\leq\epsilon<1\) | Four vertices | Four isolated points |
| \(1\leq\epsilon<\sqrt2\) | Vertices and the four side edges | One square loop |
| \(\epsilon\geq\sqrt2\) | All subsets of the four vertices | A filled tetrahedron |
At the middle scale there are no triangles: every three square vertices include a pair separated by a diagonal. At the last scale, both diagonals, all four triangles, and the tetrahedron enter together. It is a full \(3\)-simplex, rather than just a square with its planar interior colored in.
A tetrahedron’s surface would include its four triangular faces but omit its \(3\)-simplex. That surface encloses a cavity; the full tetrahedron does not. Listing simplices carefully saves us from letting a drawing invent a hole.
3.7 The role of scale
A small scale leaves isolated points. A sufficiently large scale turns a finite metric space into one full simplex. Neither extreme usually describes the structure we hoped to study.
There need not be a single best scale. Persistent homology will track the changes across scales rather than requiring us to choose one at the start. We will still need to choose a metric, preprocessing, and a computational limit; persistence does not make those choices disappear. The uncertainty chapter explains how to examine their consequences.
3.8 Stop and predict
- At \(\epsilon=1.2\), does our square Rips complex contain \(\{a,b,c\}\)? Does it contain \(\{a,b\}\)?
It contains \(\{a,b\}\) because \(d(a,b)=1\leq1.2\). It does not contain \(\{a,b,c\}\) because \(d(a,c)=\sqrt2>1.2\). One missing edge is enough to exclude a simplex.
- Three points form an equilateral triangle of side length \(1\). Does their Rips filtration ever contain an unfilled triangular loop for a positive range of scales?
No. Below \(1\), there are only vertices. At \(1\), all three edges and the \(2\)-simplex enter together. A hand-drawn triangular graph has a loop, but its Rips completion fills that loop immediately. Any interval from an artificial ordering within this tied scale has zero length.
- What must you omit from a full tetrahedron to leave a hollow sphere? May you instead omit one edge but retain every triangular face?
Omit the \(3\)-simplex and keep its four faces, edges, and vertices. The realization is the tetrahedron’s surface, homeomorphic to a sphere. Keeping a triangle while deleting one of its edges violates closure under faces, so it would not be a simplicial complex.
I invented the laziness. The bridge problem is real.↩︎