The Basics of Homology Theory

Algebraic topology is a field that seeks to establish correspondences between algebraic structures and topological characteristics. It then uses results from algebra to infer and uncover results about topology. It's a pretty powerful method.

Roughly speaking, AT provides two different frameworks for characterizing topological spaces. These are homotopy and homology. In this post, we'll start to take a look at homology, which differs from homotopy in that it's less powerful in some senses, but significantly easier to work with and compute which endows it with a different sort of power.

Some Definitions to Start

Definition Say we're working in $\mathbb{R}^n$. The $p$-simplex, defined for $p \leq n$, is $$\Delta_p := \left\{x = \sum_{i=0}^p \lambda_i e_i \mid \sum_{i=0}^p \lambda_i\ = 1, \lambda_i \geq 0\right\}$$ $\Delta_p$ is a generalization of the triangle to $p$ dimensions. Now, there are actually two types of homology which have developed in the annals of mathematics. The first of these is singular homology, which concerns itself with the study of topological spaces via the mapping of simplices into these spaces. The other type is called Cech homology which handles the study of topology via the approximation of topological spaces with spaces of a certain class, namely those which admit a triangulation. Of these two branches of homology, singular is by far the more prevalent in the literature and is the one we'll delve into here.

Given a triangular polytope like we've defined in $\Delta_p$, one operation we might like to consider is using that region as a way to sort of define regions of interest around not just basis vectors, but any arbitrary collection of vectors in the space. Given such a set $\{v_0, \ldots, v_p\}$ we can denote by $[v_0, \ldots, v_p]$ the mapping of $\Delta_p \to \mathbb{R}^n$ defined by $$\sum_{i=0}^p \lambda_i e_i \to \sum_{i=0}^p \lambda_i v_i$$ What this gives us from an intuitive perspective is the simplex expanded or shrunken to cover the span of the $v_i$. One nice property of this map is that its image is convex. We call the resulting simplex the affine p-simplex, and we sometimes refer to the $\lambda_i$ in this context as barycentric coordinates.

In addition to mapping between simplices and sets of vectors, we'd like to define a way to map between a $p$-simplex and a ($p + 1$)-simplex and vice versa. The mapping from $p$ to $p + 1$ is called the $i$th face map and is notated as $$F_i^{p+1} : \Delta_p \to \Delta_{p + 1}$$ It is formed by deleting the $i$th vertex in dimension $p + 1$. To notate this, we can write $[e_0, \ldots, \hat{e_i} , \ldots, e_{p+1}]$ where the hat indicates that $e_i$ is omitted. This face map is so named because it embeds the $p$-simplex in the $p+1$-simplex as the face opposite $e_i$, the vertex that's being deleted.

Definition For a topological space $X$, a singular $p$-simplex of $X$ is simply a continuous function $\sigma_p : \Delta_p \to X$.

Basically, our goal in defining the singular $p$-simplices is to provide a sort of "basis" for the triangulation of an arbitrary topological space. In this way, the singular $p$-simplex gives us a way to cover some patch of $X$ with a triangle-like region. Accordingly, we can define a group which in some sense acts like a vector space of these triangular basis regions. By allowing the linear combination of maps on these elements, we can give entire triangulations of the space $X$.

Definition The singular p-chain group $\Delta_p(X)$ is the free abelian group that's generated by the singular $p$-simplices.

In more concrete terms, the elements of $\Delta_p(X)$ are called $p$-chains and are simply linear combinations $$c = \sum_\sigma n_\sigma \sigma$$ of $p$-simplices with coefficients $n_\sigma$ coming from some ring (usually the integers).

We can recover how the singular $p$-simplex maps the faces of $\Delta_p$ to $X$ by simply composing the map $\sigma$ with the face map $F_i^p$. This is called the ith face of $\sigma$ and is written $$\sigma^{(i)} = \sigma \circ F_i^p$$ For a given singular $p$-simplex $\sigma$, we can defined a $(p-1)$-chain that gives the boundary of $\sigma$ as $$\partial_p \sigma = \sum_{i=0}^p (-1)^i \sigma_p^{(i)}$$ The boundary operator extends to chains in the natural way, by distributing over addition, $\partial_p c = \partial_p (\sum_\sigma n_\sigma \sigma) = \sum_\sigma n_\sigma \partial_p \sigma$. This law, in fact, makes $\partial_p$ into a homomorphism of groups $$\partial_p : \Delta_p(X) \to \Delta_{p-1}(X)$$ Now, we introduce a key fact about the boundary operator that will allow us to define homology groups. You should be well aware of the saying "the enemy of my enemy is my friend". Well, in algebraic topology boundaries are our enemies and we have a similar statement, "the boundary of my boundary is a loser (zero)". In other words, for any $\sigma$, $$\partial_p(\partial_{p + 1} \sigma) = 0$$ I'll skip the proof for this because it's just a really nasty rearrangement of a complicated sum, but if you're truly interested you can find it in pretty much any algebraic topology textbook.

We call $im(\partial_{p+1}) = B_{p}(X)$ the boundary group and $ker(\partial_p) = Z_p(X)$ the cycle group. Note that the above identity implies that $im(\partial_{p + 1})$ is a subgroup of $ker(\partial_p)$ which is equivalent to saying $B_p(C) \leq Z_p(C)$.

The Homology Group

We define the $p$th homology group as the quotient $$H_p(X) := Z_p(X) / B_p(X)$$ In other words, this is the group of cycles modded out by the group of boundaries. To give a rough intuition, the rank of $H_p(X)$ tells us the number of $p$-dimensional "holes" contained in $X$. You can think about it as follows. If each cycle in $X$ is equivalent to some boundary, then those boundaries have no "interior" excisions. However, if a hole does exist, then there will be some discrepancy between boundaries and cycles, and the number of such discrepancies (holes) will be given by the rank of $H_p(X)$.