homology.qmd

# Simplicial homology

> "Her thoughts were theorems, her words a problem, \
> As if she deem'd that mystery would ennoble 'em." \
> --- Lord Byron, in "Don Juan"

Simplicial complexes are excellent tools to model finite spaces.

## What is a hole?

How can we detect a hole in a simplicial complex? Intuitively, a hole is something *that is not there*. In a circle, we have a boundary but have no "filling".

Let's reason using the following simplicial complex $S$ (with vertex set $V = \{a, b, c \}$ ) as an example:

![An ugly simplicial complex homeomorphich to a circle.](images/hm-circle.png)

Let's add a direction to the 1-simplices of $S$ as follows:

![$K$ with an ordering on its 1-simpleces.](images/hm-circle2.png)

Intuitively, a 1-dimensional hole is formed when we have a sequence of 1-simplices $a \to b \to c$ where the last element connect with the first, like a snake biting its own tail. It would be nice if we could detect these kind of *cycles* in a formal manner... Luckily for us, [Poincaré was a very smart guy](https://en.wikipedia.org/wiki/Analysis_Situs_(paper)).

We will now try to define an algebra on $S$ that allows us to detect cycles and holes.

To describe a segment in a vector space, we need only two points: the beggining $u$ of the segment and its end $v$. This vector can then be written as $v-u$. Inspired by this, we define a function $\delta$ called *boundary operator* on the edges of $S$ by 

$$
\delta([x, y]) = y - x.
$$ 

But what is the meaning of $a - b$? This is just a "formal sum": we treat $a, b, c$ as a basis of a linear space. This vector space is called a [free group](https://en.wikipedia.org/wiki/Free_abelian_group) on $V$. We can also consider the same formal sums of 1-simplices, and extend $\delta$ linearly, ie, 

$$
\delta(e_1 + \lambda \cdot e_2) = \delta(e_1) + \lambda \cdot \delta(e_2)
$$

for any $e_1, e_2$ 1-simplices of $S$. These vector spaces generated by the formal sums of $S_k$ are denoted $C_k(S)$ and called the *simplicial k-chains*, for k = 0, 1. We also impose that $[a, b] = -[b, a]$ (this is necessary to independ on the orientation).

Now, if we take the sum of the edges $[a, b], [b, c]$ and $[c, a]$, we have

$$
\delta([a, b] + [b, c] + [c, a]) = (b - a) + (c - b) + (c - a) = 0
$$

which means that a cycle has a trivial boundary. Define 

$$
aaa
$$

A hole is a boundary without it

homology as a tool to count holes; describe the algebra intuitively, formalise later