Heroes of topology

Atiyah & Hirzebruch

Atiyah & Hirzebruch

On Friday I attended the Spitalfields Day at the Royal Society of Edinburgh, where I got to meet two of the greatest living topologists: Professor Sir Michael Atiyah and Professor Friedrich (Fritz) Hirzebruch.  The purpose of the day was to celebrate Professor Hirzebruch’s election to an Honorary Fellowship at the RSE, and it involved three talks about the history of topology and the particular contributions of these two men.

I really have to impress on you how privileged I feel to have met these men, since if you are not a mathematician you probably won’t have heard of them.  It is like a physicist getting to meet Stephen Hawking and Ed Witten, or a biologist getting to meet James Watson and Richard Dawkins. Hmm…Not sure if those are the best analogies but I hope you get the point!

Hirzebruch was born in the (now German, but formerly Prussian) city of Hamm in 1927, with Atiyah making an appearance in London two years later.  They met at the Institute for Advanced Study in Princeton in 1955 and have since remained lifelong friends and collaborators.  After Princeton, Atiyah came home to England and developed his career at Oxford and Cambridge, while Hirzebruch became a professor at Bonn.

In fact, there was another professor of the same age starting in Bonn at about the same time, whose name you may recognise: Joseph Ratzinger.  Sadly he was based in the department of theology, not of mathematics, so he and Hirzebruch never officially met (though they probably shared a room at some point!).  But by one of those amazing coincidences they both happened to be in Scotland on Friday: Ratzinger as head of the Catholic Church and Hirzebruch as head of the topological church.  One can only imagine what the world would be like if Hirzebruch had met Ratzinger in Bonn and converted him to the religion of mathematics.

So, what great things did Atiyah and Hirzebruch prove together?  The two big ideas which revolutionised topology were K-theory and Index Theory. [At this point, I’m going to have to come clean and admit that I know very little about these subjects, mainly because they are so very difficult (and I have been too lazy to learn them).  Thankfully, there are some good introductions, one of them being Atiyah’s Wikipedia page and another being Allan Hatcher’s textbook.  I will try to summarise these now.]

Picture a circle.  At every point on this circle I want you to picture a line coming out of it at right angles.

Line bundle

Adding in lines

Annulus

Finished annulus

Got something like this?  The shape is called an annulus, and it’s an example of a line bundle over the circle.  Are there any others?  Well, if the circle sits in 3D space instead of on our paper, then the answer is ‘yes’.  Each line, after it is joined to the circle, is free to rotate through 360°.  Even if we insist that the angles of the lines should change continuously as we move around the circle, this still gives us lots of room for manoevre.

Möbius strip

The most famous example of a non-trivial line bundle is the Möbius strip.  And this is the only other example of a line bundle over the circle.  If, for example, you let the lines trace out two twists in the surface, this turns out to be topologically indistinguishable from the un-twisted strip.

We can play this game with any geometric object: putting a line over every point.  Then we can think of those lines as being 1-dimensional vector spaces (think of each line as being the number line between 0 and 1, so you can add and multiply together points). We can then generalise this, so that instead of lines we have n-dimensional vector spaces over each point.  This is called a vector bundle.    Just as with the circle, we’d like to know how many different vector bundles of a particular dimension we can have for the object we picked.  As you may have guessed by now, this has turned out to be an extremely difficult problem.

Atiyah and Hirzebruch showed that there is a way of seeing two vector bundles as being the same which turns classes of vector bundles into a group, called the K-group.  A group is a powerful mathematical tool, developed to study symmetry, and the structure of the K- group provided very deep information about the object being studied.  K-theory is such a fundamental concept that it made mincemeat out of other theorems and proofs, which were originally thought to be very difficult but which turned out to be special cases of K-theory.

Ok, now on to Index Theory.  Once again you start with a topological object, like a Möbius strip or a sphere, but this time you have rules for how it curves.   The ‘rule’ is called a differential operator.  It’s like being on a mountain, and the differential operator telling you how steep the slope is in any direction.  The problem is to find out where the flat bits of the mountain are; that is, where the differential operator vanishes.  There is a (whole) number, called the analytical index, which measures how many flat bits there are.

What Atiyah, and another guy called Singer, did was to prove that this index could be computed from the topological data of the object you started out with.  For example, on an orientable surface like the torus, the index of a particular operator turns out to be the Euler characteristic.  These ideas had a huge impact in all areas of mathematics, not just topology, and also in physics.

Sir Michael is still working hard in his 80s, trying to find new ways to apply topology to physics.  One of the ideas he put forward to us at the RSE on Friday was that physical invariants, such as electric charge and the baryon number, could be the indices of a topological object.  It remains only to work out which topological objects (and which differential operators) are associated with basic particles such as protons and neutrons, and why!

It certainly won’t surprise me to find that topology is at the heart of the most basic elements of the universe, and that Atiyah and Hirzebruch’s deep theorems are the key to figuring it all out.

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