Introducing homology

A lot of things have happened since my last post, and I’ve been waiting for a great way to follow Anna’s fantastic series of SIAGA posts.

On February 16th Professor Robert Ghrist from the University of Pennsylvania gave the annual Potter lecture at the University of Aberdeen. The Potter lecture is to be aimed at a general audience, and his title was “Putting Topology to Work”. He discussed applications of topology to various areas of engineering and science and his talk included a great introduction to a topological invariant called homology.

Topologists work with homology a LOT. It has appeared in the title of my mathematics dissertation at undergrad, essay at masters level and I am pretty sure my PhD thesis title (touch wood) will also contain it. However I have never been good at explaining what homology is in layman’s terms (despite many attempts), so Professor Ghrist’s lecture was particularly inspirational.

A couple of weeks ago I gave a short talk at a London Mathematical Society Women in Mathematics day and tried to give a better description of homology than I have done before. There are some pictures involved so I thought I would recreate that section of my talk here. I’ll screenshot some of my slides and also add text and some extra sketches.

top spaces to sequence of groups

Homology is a process where we start with a topological space X and associate to it a sequence of abelian groups called homology groups, and denoted H_*(X) where * is a natural number (0, 1, 2, 3, …). Some examples of topological spaces are spheres, surfaces and manifolds (which are higher dimensional analogues of surfaces i.e. I can’t draw them).

top spaces

 

h0h1h2

 

So what do these groups tell us? H_0(X) tells us about the connected components of our space. If the space is one point, the rank of H_0(X) will be 1, if it is a circle the rank of H_0(X) will still be 1 but if it is two disjoint points or circles the rank of H_0(X) will be 2 and so on.

 

H0

 

H_1(X) tells us in some sense about ‘holes which look like a circle’. So it will let us know that a circle has one ‘hole that looks like a circle’, a figure of 8 has two, etcetera.

Similarly H_2(X) tells us about ‘holes which look like a 2- sphere’, in the sense that you can blow up a beach ball and what you get is a 2- sphere, so H_2(X) will tell you there is a hole in your beach ball which ‘looks like a 2-sphere hole’. You can also blow up a rubber ring or inner tube and in the same sense, H_2(X) will tell us these torus surfaces have ‘holes which look like 2-spheres’ or ‘holes which look like beach ball holes’. We can’t really visualise what the homology tells us after H_2(X), since it tells us about holes in higher dimensions than 2.

 

beachball rubber ring

 

So why do we do this? We might want to know something about a topological space, but maybe we can’t simply draw the space as it lives in a very high dimension. But the homology of a space is a sequence of groups which tells us about holes of all dimensions: and we know lots about groups! We can try to work out what the homology groups of a space are, we can do things such as study maps between these groups, and there is generally a lot more structure in the sequence of groups for us to take advantage of. So by looking at homology we can learn things about a space that we cannot draw or visualise.

Homology is also functorial in the sense that if we have two spaces X and Y and a map between them (the downwards black arrow in the diagram below), we can look at the homology of X and the homology of Y (horizontal wiggly arrows) and the map between X and Y will induce  a map between the homologies (the dotted green arrow). So because we know a lot about maps between groups this can tell us something about the possible maps between X and Y.

functorial

In my talk I was focusing on the homology of a group rather than that of a space, so how do we do that? Well we start off with a group and we associate to it something called a classifying space (see my previous post for an example). Calculating the homology of this space is then the same as calculating the homology of the group.

group 2

I also used homology with different coefficients, such as \mathbb{Z}_2 instead of the usual integer coefficients \mathbb{Z}. This allows us to manipulate what sort of abelian groups we get when taking homology, for instance using \mathbb{Q} coefficients will give us a \mathbb{Q}-vector space. Sometimes we do this to make our problem easier to solve, or sometimes the problem itself prescribes that we use different coefficients.

coeffiecients

So now I have told you about homology, next time I will follow up with a post on the hot topic of homological stability!

And to reward you for reading to the end, here is a great comic drawn by my friend Tom!

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6 thoughts on “Introducing homology

  1. confused May 6, 2016 / 4:36 am

    Something confusing: you start w/ an (abelian) group. But then talk about rank and coefficients so somewhere a vector space is being used. Where does that enter the picture?

    Like

  2. Rachael May 6, 2016 / 8:41 am

    Thank you for your comment!

    So first of all I talk about rank of an abelian group. Suppose our coefficients are the integers, then what I mean by this is that if we have a connected space, we will have H_0(X, \mathbb{Z})=\mathbb{Z} but if our space consists of two connected components we will have H_0(X, \mathbb{Z})=\mathbb{Z} \oplus \mathbb{Z} where \oplus means the direct sum. This is still an abelian group, where the elements can be written as (a,b) with a, b in \mathbb{Z} and the group operation is given by applying the operation on each \mathbb{Z} seperately i.e. (a,b) + (c,d) = (a+b, c+d). So by rank, in this case I just mean how many copies of the integers there are.

    Then when I change coefficients I say we are now getting a vector space over the coefficient group. If you look at the axioms of a vector space, and forget the ones relating to scalar multiplication you have exactly those for an abelian group (with operation +) and so if I have a vector space then I still have an abelian group, just with some more structure (the scalar multiplication stuff).

    I hope you are less confused now, please let me know if there is anything more I can/should say!

    Liked by 1 person

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