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Homology describes the shape of a space in a really amazing way. At first it seems strange and confusing, but after a careful study, it comes implicit in a lot of mathematical phenomena. For instance, in the Euler-Poincaré Characteristic: we take some space, triangulate it (break it up in "small pieces") and compute the alternate sum of the number of pieces of each dimension:

$$\chi(X)=\sum_i (-1)^i c_i(X)$$

where $c_i(X)$ is the number of $i$-dimensional "faces" of $X$. Surprisingly enough, it does not depend on the chosen triangulation

$X$$c_0$ vertices$c_1$ edges$c_2$ faces$\chi=c_0-c_1+c_2$
1--1
43-1
76-
55-0
99-
5511
991
6105
81262
6128
123020
3264320
367236


However, this characteristic is also expressed in homological terms (with no need of triangulation), and this is the proper way indeed. So homology effectively describes the shape of a topological space

$$\chi(X)=\sum_i (-1)^i \text{rk }H_i(X)=\sum_i (-1)^i b_i(X)$$

where $b_i=\text{rk }H_i(X)$ are the Betti numbers

$X$$b_0$$b_1$$b_2$$\chi=b_0-b_1+b_2$
1--1
10-1
11-0
1001
1012
1210



Our goal is to understand the shape of a space. How shall we distinguish a sphere from a torus? Homology has a clear motto:

FIND SOMETHING WITHOUT BOUNDARY, THAT IS NOT THE BOUNDARY OF ANYTHING



and... it works! Of course, objects with boundary (e.g. open curves) aren't very useful, they may move wildly



but objects without boundary... closed objects... these may be more meaningful



In the sphere, a curve without boundary is always the boundary of something... but in the torus we have a curve without boundary that is not the boundary of anything. This is a true difference! It seems to work!



Generally speaking, these objects without boundary that are not the boundary of anything catch "holes" in different dimensions... the holes of dimension 0 are unconnected components, the holes of dimension 1 may be surrounded by a loop, whereas those of dimension 2 may be enclosed by surfaces...



But one could argue, "isn't the loop in the torus the boundary of the rest of the torus?"



Well... we're talking about the boundary without knowing exactly what it is... So let's spend some time to define the boundary operator $\partial$


To define the boundary operator $\partial$, we'll proceed one step at a time:

  1. $\partial$ maps objects of dimension $k$ to objects of dimension $k-1$, always one dimension below



    Objects of dimension 0 are points, those of dimension 1 are curves, those of dimension 2 are surfaces... The set of "objects of dimension $k$" will be denoted $C_k(X)$, so we have

    $$\partial:C_k(X)\longrightarrow C_{k-1}(X)$$

    for every $k\geqslant 0$ (the boundary of a point is nothing, or more algebraically stated, is 0)

  2. The boundary of something has no boundary (is 0). We may convince ourselves with the examples above. Algebraically, we express it like this

    $$\partial^2=0$$

    and is such an important and ubicuous property in mathematics that gives rise to the whole field of Homological Algebra

  3. For this to work well, all the objects must be oriented. The very concept of orientation is very subtle; for now it will be some additional property that takes two opposite values ("one the negative of the other one")

    • Points will be labelled as $+$ or $-$
    • Curves have a direction
    • Surfaces are painted blue in one face and yellow in the other face; blue denotes counterclockwise rotation and yellow denotes clockwise rotation
    • Volumes have some sort of "helicity"...


    and this, of course, is taken into account by the boundary operator! A curve starts in a $-$ point and ends in a $+$ point; the spin in the surfaces must match the direction of the boundary curve...



    Why is this a key fact for the boundary operator? Because this way $\partial$ works perfect when dividing objects



    Great! Equal objects with different orientation act as opposite and cancel out. So everything works with a marvelous algebraic taste. Moreover, this clarifies the problem we had with the torus: if we cut the torus, each edge of the surface adds a boundary, but since they have opposite orientations, they cancel out!



    So it is true: the beforementioned loop is not the boundary of anything



So we should find objects without boundary that are not the boundary of something. We'll consider three groups:

  • $C$: objetcs (with or without boundary)
  • $Z$: objetcs without boundary
  • $B$: objetcs without boundary that are also the boundary of something


And the inclusion $B\subset Z\subset C$ holds. Apparently, the set $Z\smallsetminus B$ is the interesting one. But... is this the best approach, the most "mathematical"?

Let's think for a moment in the whole numbers $\mathbb{Z}$. Among the whole numbers there is a "phenomenon", that of being multiple of $5$. This phenomenon is interesting precisely because the are whole numbers that are not multiple of $5$. We may think of it as if $Z=\mathbb{Z}$ and $B=5\mathbb{Z}$ the multiples of $5$. One may show the fact that there are numbers that are not multiples of $5$ examining the set

$$Z\smallsetminus B=\{\cdots,-6,-4,-3,-2,-1,1,2,3,4,6,7,8,9,11,12,13,14,\cdots\}$$

But for a mathematician, working with this set is very annoying. It isn't even closed for the sum ($2$ and $3$ are in $Z\smallsetminus B$ but $2+3=5\not\in Z\smallsetminus B$), there is no interesting structure for this set. So let's consider a totally different approach

$1$ and $6$ are both non-multiples of $5$, and are distinct, but it could be said that both express the same not-being-multiple-of-$5$, because both are of type $5k+1$. Perhaps we could abstract the number and consider the class, so we'll say

$$[1]=[6]\qquad\text{ because }\qquad 1-6\in B=5\mathbb{Z}$$

In the very same spirit, we define

$$[a]=[b]\qquad\text{ if }\qquad a-b\in B=5\mathbb{Z}$$

This gives rise to five classes, making up $\mathbb{Z}_5=\{[0],[1],[2],[3],[4]\}$, which happens to be extremely rich from a structural point of view: not only is it closed for the sum and product modulus $5$, which constitutes the set as a group and a ring, but it is also a finite field. And this procedure is standard in mathematics: it is the quotient $Z/B$.

So the right point of view, also for the homology, is not substraction $Z\smallsetminus B$, but quotient $Z/B$. Some examples will corroborate how useful is this procedure


Let $C_k(X)$ be the set of objects of dimension $k$ inside $X$. Since we are interested in an algebraic approach, we'll work with (formal) linear combinations that we may add and substact easily. That is, elements of $C_k(X)$ may be of the form



where $\lambda_i$, $\mu_j$, $\eta_k$ are whole numbers. These are called chains. We keep in mind that a negative "amount" of objects is equivalent to a change of orientation, as discussed above: swapping $-$ and $+$, inverting the direction of a curve, interchanging blue and yellow... So the boundary operator $\partial$ is better seen as a linear operator

$$\partial_k:C_k(X)\longrightarrow C_{k-1}(X)$$ which satisfies $\partial_{k-1}\circ\partial_k=0$ (or more succinctly $\partial^2=0$). This implies

$$\text{im }\partial_{k+1} \subset\text{ker }\partial_k$$

and it makes sense to define

$$Z_k(X)=\text{ker }\partial_k$$ $$B_k(X)=\text{im }\partial_{k+1}$$

(called cycles and boundaries respectively) and yes, $B_k(X)\subset Z_k(X)\subset C_k(X)$. And in this context, we define the homology groups

$$H_k(M)=\dfrac{\text{ker }\partial_k}{\text{im }\partial_{k+1}}=\dfrac{Z_k(M)}{B_k(M)}$$

Strange? Somewhat. But extremely useful. The homology groups are made up by homology classes, that merge homologous cycles, that is, cycles whose difference is a boundary. And, as a matter of fact, homologous cycles, those whose difference is a boundary, are precisely those that catch the same "hole". Astonishing, isn't it? So astonishing that has fascinated and keeps fascinating all sort of scientists over decades




Let's explore some concrete examples of homology. The simplest homology to be computed is homology of order zero, which detects the number of (path) connected components in the space. Why? For $k=0$, each element of $C_0(X)$ is just an integer-valued sum of points in $X$, for instance $\sigma=7p-3q$, $p$, $q\in X$. Points have no boundary, so $\partial_0:C_0(X)\longrightarrow 0$ is the null map and $Z_0(X)=C_0(X)$. On the other hand, $\partial_1:C_1(X)\longrightarrow C_0(X)$ maps each open path to a pair of points, one point with a $+$ sign and the other one with a $-$ sign. And any such pair of points belongs to the image of $\partial_1$ as long as there is some path in $X$ from the $-$ point to the $+$, that is, as long as they're in the same connected component! As a consequence, in $H_0(X)=Z_0(X)/B_0(X)$ we're free to move a point inside its component, as shown in the picture above, and we may also calcel out $+$ and $-$ points in the same component. This being so, each homological class in $H_0(X)$ assigns a net number of points to each component, and thus $H_0(X)\simeq \mathbb{Z}^n$, where $n$ is the number of connected components of the space. Fantastic, isn't it?





In the very same spirit, $H_1(X)$ detects holes surrounded by a loop. The quotient ensures that different loops enclosing the same hole belong to the same class





Now we can reason the homology of simple spaces

$X$$H_0(X)$$H_1(X)$$H_2(X)$
$\mathbb{Z}$--
$\mathbb{Z}$0-
$\mathbb{Z}$$\mathbb{Z}$-
$\mathbb{Z}$00
$\mathbb{Z}$0$\mathbb{Z}$
$\mathbb{Z}$$\mathbb{Z}^2$$\mathbb{Z}$



Even in the simplest spaces, the sets $C_k(X)$, $Z_k(X)$, $B_k(X)$ are usually extremely big. Think, for instance, that $C_1(X)$ contains all functions $f:[0,1]\longrightarrow X$ - that's huge. So there is a "lighter" version of homology, called simplicial homology (which will be marked with a $\Delta$), that works for triangulated spaces, and that compute the very same homology groups (i.e. isomorphic). The idea is simple: use only objects of the triangulation. So $C^\Delta_0(X)$ consists of integer-valued combinations of vertices, $C^\Delta_1(X)$ consists of integer-valued combinations of oriented edges, and so on. Everything else is the same and works fine:

$$\partial^\Delta_k:C^\Delta_k(X)\longrightarrow C^\Delta_{k-1}(X)$$ $$Z^\Delta_k(X)=\text{ker }\partial^\Delta_k$$ $$B^\Delta_k(X)=\text{im }\partial^\Delta_{k+1}$$ $$H^\Delta_k(M)=\dfrac{Z^\Delta_k(M)}{B^\Delta_k(M)}$$

But now we're only working with finite dimensional spaces (provided that the triangulation is finite) which behave much better. We may compute the simplicial homology groups. But since a space has many triangulations... we would like to have an "absolute" homology, which is the one described above. It is the singular homology, and it is one of the finest constructions in mathematics

And all this digression comes back to the Euler-Poincaré characteristic. How? Since we are now working with finite dimensional vector spaces (well, they're not vector spaces, because we're working over $\mathbb{Z}$, but it doesn't matter - we'll use rank $\text{rk}$ as the analogous of $\text{dim}$), usual linear algebra applies. In the one hand,

$$H_k=Z_k/B_k\Longrightarrow \text{rk }H_k=\text{rk }Z_k-\text{rk }B_k$$

but on the other hand,

$$B_{k-1}=\text{im }\partial_k\simeq C_k/\text{ker }\partial_k=C_k/Z_k$$ $$\text{rk }B_{k-1}=\text{rk }C_k-\text{rk }Z_k$$

and this necessarily holds along the whole chain





Plugging this into the Euler-Poincaré characteristic, we have

$$ \begin{array}{rcl} \chi & = & \sum_i (-1)^i c_i \\ & = & \sum_i (-1)^i \text{rk }C_i \\ & = & \sum_i (-1)^i (\text{rk }Z_i + \text{rk }B_{i-1}) \\ & & \text{[index trick: $\sum_i (-1)^i\text{rk }B_{i-1} = -\sum_i (-1)^i\text{rk }B_i$]} \\ & = & \sum_i (-1)^i (\text{rk }Z_i - \text{rk }B_i) \\ & = & \sum_i (-1)^i \text{rk }H_i \\ \end{array} $$

The result above is very deep: the quantities $c_i$ depend on the specific triangulation of the space $X$, but the Euler-Poincaré characteristic $\chi$ does not depend on the triangulation, and therefore constitutes a topological invariant, because it may be expressed in terms of homology. And as a side note, it shows that homology, no matter how strange seemed at first, is of crucial importance in algebraic topology