Simplicial Homology

I started writing a post about Homology Algorithms but realized that some discussion about homology should come first. And for those of you who already know what homology is, we should at least be on the same page in terms of language and notation. Besides, there’s more than one way to think about homology.

The way that I understand homology—the way I describe it here—is influenced almost entirely by Carsten Lange. He helped me understand the basics of algebraic topology so that I can read about discrete Morse theory without feeling like a total oaf. This also means that we are thinking about homology with the eventual goal of computing (i.e., with a computer) the homology of some explicit, finite discrete topological space. In particular, we will work with simplicial complexes.

References used include the classics, Munkres and Hatcher, and for a speedy introduction, Zomorodian’s “Topology for Computing”. (And, as usual, a ton of help has been provided by my advisors Frank and Bruno.)

And before I begin, let me just say: Big ups to Carsten and all dedicated educators like him who, for countless hours, over many days and weeks, sit together with students one-on-one until they have the confidence to seek knowledge on their own.

Homology in topology

The big question in topology is whether or not two spaces are homeomorphic. To say that two spaces are homeomorphic, we need only find any homeomorphism between them. But to say that two spaces are not homeomorphic, we would have to show that no function between them is a homeomorphism. One way to handle this latter (infinite) case is by using topological invariants.

A topological invariant is a property of a space that can be used to distinguish topological spaces. For example, say we have two sacks labeled $A$ and $B$. We can’t see inside them, but we can tell that sack $A$ has one item in it while sack $B$ has two. Clearly the contents of sack $A$ and sack $B$ cannot be the same.

However, if $A$ and $B$ both have one item each, we can’t tell whether those two items are of the same type. Perhaps one is a steak and the other a rice crispy treat—clearly very different objects, both nutritionally and topologically. (Can you tell I missed lunch?)

The topological invariant described in these examples is connectedness, which is only useful when the number of connected components is different. But if they have the same number of connected components, we can’t say definitively whether the objects are of the same topological type. And that’s how it works with all topological invariants.

Definition (topological invariant) Given two topological spaces $K_1$ and $K_2$, a map $\tau$ is a topological invariant if whenever $\tau(K_1) \neq \tau(K_2)$, then $K_1 \ncong K_2$. But if $\tau(K_1) = \tau(K_2)$, then $K_1 \overset{?}{\cong} K_2$ and we don’t know whether they are homeomorphic or not.

Homology groups are topological invariants. We take some geometric information about the topological space, form some algebraic object (specifically, a group) containing that information, then apply some machinery from group theory to say something about the original space.

And if two topological spaces give rise to homology groups that are not isomorphic, then those two spaces are not homeomorphic.

Definition of Homology

Let’s jump right in with a definition, then explain it in detail using an example.

Definition (homology group) For a topological space $K$, we associate a chain complex $\mathcal{C}(K)$ that encodes information about $K$. This $\mathcal{C}(K)$ is a sequence of free abelian groups $C_p$ connected by homomorphisms $\partial_p: C_p \to C_{p-1}$. \begin{align*}0 \overset{\partial_{d+1}}{\longrightarrow} C_d \overset{\partial_{d}}{\longrightarrow} C_{d-1} \overset{\partial_{d-1}}{\longrightarrow} \cdots \overset{\partial_{2}}{\longrightarrow} C_1 \overset{\partial_{1}}{\longrightarrow} C_0 \overset{\partial_{0}}{\longrightarrow} 0\end{align*} Then the $p$-th homology group is defined as \begin{align*}H_p(K):= \text{ker}(\partial_p) / \text{im}(\partial_{p+1}).\end{align*}

That’s a lot of terms we have yet to define.

Let’s start at the beginning with our input space $K$. We can assume that $K$ is a simplicial complex since we’ll only be dealing with simplicial complexes and they’re just nice to handle. So $K$ is a simplicial complex of dimension $d$. This means that for any face $\sigma \in K$, its dimension is at most $|\sigma| \le d$.

Example: simplicial complex $K$ with vertices labeled 1,2,3,4.
Figure 1: Simplicial complex $K$ with vertices labeled 1,2,3,4.

In Figure 1, we have our simplicial complex $K$. I’ve labeled the vertices so it’s easier for me to refer to the many parts of $K$. Our complex $K$ consists of

  • one 2-cell: $f_1 = [2 \, 3\, 4]$;
  • five 1-cells: $e_1=[1 \, 2], e_2=[1 \, 4], e_3=[2 \, 3], e_4=[2 \, 4], e_5=[3 \, 4]$; and
  • four 0-cells: $v_1=[1], v_2=[2], v_3=[3], v_4=[4]$.

This complex $K$ is of dimension $d=2$.

The group $C_p$ is a group of something called $p$-chains. Recall that a group is a set with a binary operation (in our case, addition) that is (i) associative, (ii) has an identity element, and (iii) has inverses. And when we refer to any group, instead of listing all the elements of the group all the time, it’s much more economical to talk about groups with respect to its generators. For the generators of $C_p$, we have to go back to $K$.

The $p$ in $C_p$ refers to the dimension of cells in $K$. A 1-chain of $K$, for example, is some strange abstract object that is generated by all the 1-cells of $K$, that is, we “add” up a bunch of 1-cells. So $e_1 + 2 e_5$ is an example of a 1-chain. It has no obvious geometric meaning (though, apparently, some people would disagree). If it helps, you can think of $C_p$ as a vectorspace whose basis is the set of $p$-cells of $K$ and the scalars are in $\mathbb{Z}$.

Figure 2: Relabeled and oriented.
Figure 2: Relabeled and oriented.

Notice that the labeling of the vertices from Figure 1 gives us a nice built-in orientation, i.e., ordering, for each cell, see Figure 2. For example, edge $e_1 = [1 \, 2]$ so the orientation we give this edge is from $v_1$ to $v_2$ (because $1<2$). This orientation will help us define the additive inverse of a 1-cell. So $-e_1$ is what you would expect it to be; it’s just $e_1$ oriented in the opposite direction, from $v_2$ to $v_1$.

Now the definition states that $C_p$ is not just any old group. It is a free Abelian group. This is not as bad as it sounds. It just means, again, that we can just go back to thinking of $C_p$ like it’s a vectorspace.

So $C_p$ is free, which just means that every $p$-chain in $C_p$ can be uniquely written as $\sum_i n_i \sigma_i^p$, where $n_i \in \mathbb{Z}$ and $\sigma_p$ are $p$-cells. This idea should be deeply ingrained into your understanding of basis vectors in a vectorspace.

And $C_p$ is also Abelian. So our addition is commutative $\sigma + \tau = \tau + \sigma$ and that also let’s us keep thinking about this vectorspace idea.

So now that we know what the $C_p$ are, let’s talk about the homomorphisms $\partial_p$ between them. These homomorphisms have a special name; they are called boundary maps. This is where the geometry and combinatorics fit together very nicely. Let’s go back to our Figure 2.

The boundary of some $p$-cell $\sigma^p=[v_0 \, v_1 \, \dots \, v_p]$ is a $(p-1)$-chain with the alternating sum $$\partial_p(\sigma) = \sum_{i=0}^{p} (-1)^i [v_0 \, v_1 \, \cdots \, \hat{v_i} \, \cdots \, v_p ]$$ where $\hat{v_i}$ means you leave that vertex out.

For example, $\partial_1(e_1) = v_2 – v_1$ or $\partial_2(f_1) = \partial_2([2 \, 3 \, 4]) = [3 \, 4] – [2 \, 4] + [2 \, 3] = e_3 + e_5 – e_4$. This corresponds nicely with the figure. Just follow along the blue arrows on the boundary of $f_1$ from $v_2$ in a counter clockwise direction.

The boundary function works similarly for all dimensions $p$. For the higher dimensions, just keep in mind what I said earlier about orientation being determined by the labels. The orientation for an edge is easy to visualize, but not so much for,  say, a 6-simplex. Everything works out nicely combinatorially with the above definition.

And because $C_p$ is a free abelian group, we see that the boundary map $\partial_p$ is a homomorphism. Notice also that $\partial_p \circ \partial_{p+1} = 0$. Whenever this happens (to any sequence of Abelian groups with homomorphisms between them with this property), we call such a sequence a chain complex.

Geometry of homology

Let’s introduce a few more important terms.

The set $B_p = \text{im}(\partial_{p+1})$ is the set of boundaries and the set $Z_p = \text{ker}(\partial_p)$ is the set of cycles (which, in German, is “Zyklus”).

So a cycle is any $p$-chain that sums up to zero. This is a very combinatorial/algebraic definition for a word that sounds very geometric. Intuitively, when you hear the word “cycle”, you might think of some path that closes, maybe like a loop. It’s very important, however, to recognize that “cycle” and “loop” (from homotopy theory) are very very different objects.

Let me go off on a short tangent here. Have you noticed that the word “curve” is different depending on who you talk to? For a physicist, a curve is a trajectory–a function $\gamma: [0,1] \to X$ for some space $X$. For a geometer, however, a curve is a subset of a space. Merely the image $\text{im}(\gamma)$. So it doesn’t matter whether a mosquito flies around in a circle 7 times or just once; it still traces just a single circle.

Analogously, a cycle doesn’t care about how you “go around the cycle” as long as the algebraic sum of the corresponding $(p-1)$-chains in the cycle adds up to zero.

And remember: since we have an Abelian group, the order of the addition (or “direction” of the cycle) doesn’t matter at all. For a loop, however, it matters whether you go around the loop clockwise or counterclockwise.

Let’s look again at the composition $\partial_p \circ \partial_{p+1} = 0$. It says, geometrically, that the boundary of $(p+1)$-chains are cycles. So the set of boundaries $B_p$ is contained in the set of cycles $Z_p$, i.e., $B_p \subseteq Z_p$.

Not only that, because $\partial_{p+1}$ is a homomorphism, $B_p$ is even a subgroup of $Z_p$. Geometrically speaking, the quotient group $H_p = Z_p / B_p$ means we partition the set of cycles $Z_p$ to those that bound and those that don’t. This means we can isolate cycles that do not bound cells. That is, we can find holes!

Figure 2: Relabeled and oriented.
Figure 2: Relabeled and oriented.

Let’s go back to our example and think about $H_1 (K)$. The “simplest” 1-chains that make a loop are $e_1 + e_4 – e_2 = \ell_1$ and $e_3 + e_5 – e_4=\ell_2$. All other loops are some linear combination of $\ell_1, \ell_2$. For example $e_1 + e_3 + e_5 – e_2 + (e_4 – e_4) = \ell_1 + \ell_2$. So $Z_1$ is generated by $\ell_1, \ell_2$, i.e., $Z_1 = \{c\, \ell_1 + d\, \ell_2 \mid c,d \in \mathbb{Z} \} \cong \mathbb{Z}^2$. But we also know that $\ell_2 = \partial_{2} (f_1)$ and since $f_1$ is the only 2-cell in $K$, we have $C_2 = \{ c \,f_1 \mid c \in \mathbb{Z} \}$ and therefore $B_1 = \text{im}(\partial_{2}) = \{ c \, \ell_2 \mid c \in \mathbb{Z}\} \cong \mathbb{Z}$.

So $H_1 = Z_1/B_1 \cong \mathbb{Z}^2 / \mathbb{Z} \cong \mathbb{Z}$.

Using homology

Now that the homology group has been defined, we should know what it means. There is an important theorem in group theory that we should recall at this point. It’s so important that it is a fundamental theorem.

Theorem (Fundamental Theorem of Finitely Generated Abelian Groups) Every finitely generated Abelian group $G$ is isomorphic to a direct product \begin{align*}G \cong H \times T, \end{align*} where $H = \mathbb{Z} \times \cdots \times \mathbb{Z} = \mathbb{Z}^{\beta}$ is a free Abelian subgroup of $G$ having finite rank $\beta$ and $T = \mathbb{Z}_{t_1} \times \mathbb{Z}_{t_2} \times \cdots \times \mathbb{Z}_{t_k}$ consists of finite cyclic groups of rank $t_1, \dots, t_k$.

The $T$ is called the torsion subgroup of $G$; the $\beta$ is unique and called the Betti number of $G$; and the $t_1, \dots, t_k$ are also unique and called the torsion coefficients of $G$.

This fundamental theorem says that the homology group $H_p$ (which is a quotient group of a finitely generated Abelian group over a finitely generated Abelian group and therefore itself a finitely generated Abelian group) will always be isomorphic to something that looks like \begin{align*}H_p \cong \mathbb{Z}_{t_1} \times \cdots \times \mathbb{Z}_{t_k} \times \mathbb{Z}^{\beta}\end{align*} a finite direct sum of (finite and infinite) cyclic groups. So now all topological spaces can be nicely categorized by their homology groups according to their Betti numbers and torsion coefficients.

The homology groups $H_p$ are of dimension $p$ and each of them have an associated Betti number $\beta_p$. Since we figured out for our example that $H_1 (K) \cong \mathbb{Z} = \mathbb{Z}^1$, our Betti number $\beta_1 = 1$.

These Betti numbers have a wonderful geometric meaning.  This number $\beta_p$ counts the number of $p$-dimensional holes in $K$. Ah, yes. We finally get to something meaningful and even useful!

Betti numbers can count, as in our example, the number of holes.

Since $\beta_1(K)=1$, this means $K$ has one 1-dimensional hole. And there is indeed a hole in $K$ (because there is no face $[1 \, 2 \, 4]$).

But what is a “hole”?

Honestly, I don’t know how to draw a good picture of a general $p$-dimensional hole (though examples of specific cases are easy). Thinking of holes like in homotopy theory–as the empty spaces where $p$-spheres $S^p$ cannot be shrunk down to a point–is a very nice and intuitive notion. But there are some spaces that,  like the Poincare sphere for example, have $H_1 = 0$ and $\pi_1 \neq 0$. So it’s not enough to think about holes as the holes we see when working with fundamental groups.

We saw from the definition that homology groups are equivalence classes of cycles, of those that bound and those that don’t. So we collect all the “simplest” cycles and check whether they are the boundary of any faces. Any cycle that does not have a corresponding face which it bounds must bound a hole.

It turns out that counting holes has all kinds of industry applications. It can be used to analyze the structure of solid materials or proteins. It can be used in image processing or pattern recognition. It can even be used in robotics and sensor networks.

So homology algorithms have been developed to help these industry people to count holes. And now we know exactly how to write such an algorithm, right?