After the workshop on categorical groups in Barcelona, I went to Granada - the world capital of categorical groups! Pilar Carrasco, an expert on this subject, had kindly invited me to spend a week there and give some talks. Even more kindly, she put me in a hotel right next to the Alhambra.
So, this week I'll tell you about some categorical groups I saw in the Alhambra.
I've long been fascinated by that melting-pot of cultures in southern Spain called Andalusia. I wrote about it back in "week221". It was invaded by Muslims in 711 AD, and became a center for mathematics and astronomy from around 930 AD to around the 1200s, when the city of Toledo, recaptured by Catholic Spaniards, became the center of a big translation industry - translating Arabic and Hebrew texts into Latin. This was important for the transmission of ancient Greek writings into Western Europe.
The Alhambra was built after the true heyday of Andalusia, in the era when Muslims had almost been pushed out by the Catholics. Its construction was begun by Muhammed ibn Nasr, founder of the Nasrid Dynasty - the last Muslim dynasty in Spain.
In 1236, Ferdinand III of Castile captured the marvelous city of Cordoba, "ornament of the world". Ibn Nasr saw which way the wind was blowing, and arranged to pay tribute to Ferdinand and even help him take the city of Seville in return for leaving his city - Granada - alone. He started building the Alhambra in 1238. It was completed in the late 1300s.
For the mathematician, one striking thing about the Alhambra is the marvelous tile patterns. On my visit, I took photos of all the tiles I could see:
1) John Baez, Alhambra tiles, http://math.ucr.edu/home/baez/alhambra
Here are a few of my favorites:
Some people say that tilings with all 17 possible "wallpaper groups" as symmetries can be found in the Alhambra. This article rebuts that claim with all the vehemence such an academic issue deserves, saying that only 13 wallpaper groups are visible:
2) Branko Grünbaum, The emperor's new clothes: full regalia, G-string, or nothing?, with comments by Peter Hilton and Jean Pedersen, Math. Intelligencer 6 (1984), 47-56.
As mentioned in "week221", this page shows 13 of the 17:
3) Steve Edwards, Tilings from the Alhambra, http://www2.spsu.edu/math/tile/grammar/moor.htm
Of the remaining four, two seem completely absent in Islamic art - the groups called "pgg" and "pg". Both are fairly low on symmetries, so they might have been avoided for lack of visual interest.
Let me describe them, just for fun. You can learn the definition of wallpaper groups, and a lot more about them, from this rather wonderful article:
4) Wikipedia, Wallpaper group, http://en.wikipedia.org/wiki/Wallpaper_group
The group pgg is the symmetry group of this popular zig-zag method of laying bricks:
The group pg is a subgroup of pgg. If we take our zig-zag pattern of bricks and break the 180-degree rotation symmetry somehow, the remaining symmetry group is pg:
For more on tilings, try this book. Among other things, it points out that there's a lot more beauty and mathematical structure in tilings than is captured by their symmetry groups!
5) Branko Grünbaum and G. C. Shephard, Tilings and Patterns, New York, Freeman, 1987.
The mathemagician John Horton Conway has come up with a very nice proof that there are only 17 wallpaper groups. This is nicely sketched in the Wikipedia article above, but detailed here:
6) John H. Conway, The orbifold notation for surface groups, in Groups, Combinatorics and Geometry, London Math. Soc. Lecture Notes Series 165, Cambridge U. Press, Cambridge, 1990, pp. 438-447
Here's the basic idea. Take a wallpaper pattern and count two points as "the same" if they're related by a symmetry. In other words - in math jargon - take the plane and mod out by the wallpaper group. The result is a 2-dimensional "orbifold".
In a 2d manifold, every point has a little neighborhood that looks like the plane. In a 2d orbifold, every point has a little neighborhood that looks either like the plane, or the plane mod a finite group of rotations and/or reflections.
Let's see how this works for a few simple wallpaper groups.
I'll start with the most boring wallpaper group in the world, p1. If you thought pg was dull, wait until you see p1. It's the symmetry group of this wallpaper pattern:
R R R R R R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR R R R R R R R R R R R R R R R R R R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR R R R R R R R R R R R R R R R R R R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR R R R R R RThis group doesn't contain any rotations, reflections or glide reflections - I used the letter R to rule those out. It only contains translations in two directions, the bare minimum allowed by the definition of a wallpaper group.
If we take the plane and mod out by this group, all the points labelled x get counted as "the same":
R R R R R R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR R R R R R R R x R x R x R x R x R R R R R R R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR R R R R R R R x R x R x R x R x R R R R R R R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR R R R R R RSimilarly, all these points labelled y get counted as "the same":
R R R R R R RRRRRyRRRRRyRRRRRyRRRRRyRRRRRyRRR R R R R R R R R R R R R R R R R R R RRRRRyRRRRRyRRRRRyRRRRRyRRRRRyRRR R R R R R R R R R R R R R R R R R R RRRRRyRRRRRyRRRRRyRRRRRyRRRRRyRRR R R R R R RSo, when we take the plane and mod out by the group p1, we get a rectangle with its right and left edges glued together, and with its top and bottom edges glued together. This is just a torus. A torus is a 2d manifold, which is a specially dull case of a 2d orbifold.
Now let's do a slightly more interesting example:
T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T T T T T T T T T T T T T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T T T T T T T T T T T T T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T T T T T TThe letter T is more symmetrical than the letter R: you can reflect it, and it still looks the same. (If you're viewing this using some font where the letter T doesn't have this symmetry, switch fonts!) So, the symmetry group of this wallpaper pattern, called pm, is bigger than p1: it also contains reflections and glide reflections along a bunch of parallel lines. So now, all these points labelled x get counted as the same when we mod out:
T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T T T T T T T x x T x x T x x T x x T x x T T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T T T T T T T x x T x x T x x T x x T x x T T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T T T T T Tand similarly for all these points labelled y:
T T T T T T TTTyTyTTTyTyTTTyTyTTTyTyTTTyTyTTT T T T T T T T T T T T T T T T T T T TTTyTyTTTyTyTTTyTyTTTyTyTTTyTyTTT T T T T T T T T T T T T T T T T T T TTTyTyTTTyTyTTTyTyTTTyTyTTTyTyTTT T T T T T Tbut look how these points labelled z work:
T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T z T z T z T z T z T T T T T T T T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T z T z T z T z T z T T T T T T T T T T T T T TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT T z T z T z T z T z TThere are only half as many z's per rectangle, since they lie on reflection lines.
Because of this subtlety, this time when we mod out we get an orbifold that's not a manifold! It's the torus of the previous example, but now folded in half. We can draw it as half of one of the rectangles above, with the top and bottom glued together, but not the sides:
TTTT T . T . T . TTTTSo, it's a cylinder... but in a certain technical sense the points at the ends of this cylinder count as "half points": they lie on reflection lines, so they've been "folded in half".
This particular orbifold looks a lot like a 2d manifold "with boundary". That's a generalization of a 2d manifold where some points - the "boundary" points - have a neighborhood that looks like a half-plane. But 2d orbifolds can also have "cone points" and "mirror reflector" points.
What's a cone point? It's like the tip of a cone. Take a piece of paper, cut it like a pie into n equal wedges, take one wedge, and glue its edges together! This gives a cone - and the tip of this cone is a "cone point". We say it has "order n", because the angle around it is not 2π but 2π/n.
Here's a more sophisticated way to say the same thing: take a regular n-gon and mod out by its rotational symmetries, which form a group with n elements. When we're done, the point in the center is a cone point of order n.
We could also mod out by the rotation and reflection symmetries of the n-gon, which form a group with 2n elements. This is harder to visualize, but when we're done, the point in the center is a "corner reflector of order 2n".
To see one of these fancier possibilities, let's look at the orbifold coming from a wallpaper pattern with even more symmetries:
. . . . . . ................................. . . . . . . . . . . . . . . . . . . ................................. . . . . . . . . . . . . . . . . . . ................................. . . . . . .These are supposed to be rectangles, not squares. So, 90-degree rotations are not symmetries of this pattern. But, in addition to all the symmetries we had last time, now we have reflections about a bunch of horizontal lines. We get a wallpaper group called pmm.
What orbifold do we get now? It's a torus folded in half twice! That sounds scary, but it's not really. We can draw it as a quarter of one of the rectangles above:
.... . . ....Now no points on the edges are glued together. So, it's just a rectangle. The points on the edges are boundary points, and the corners are corner reflection points of order 4.
In a certain technical sense - soon to be explained - points on the edges of this rectangle count as "half points", since they lie on a reflection line and have been folded in half. But the corners count as "1/4 points", since they lie on two reflection lines, so they've been folded in half twice!
This is where it gets really cool. There's a way to define an "Euler characteristic" for orbifolds that generalizes the usual formula for 2d manifolds. And, it can be a fraction!
The usual formula says to chop our 2d manifold into polygons and compute
V - E + F
That is: the number of vertices, minus the number of edges, plus the number of faces.
In a 2d orbifold, we use the same formula, but with some modifications. First, we require that every cone point or corner reflector be one of our vertices. Then:
It turns out that if we calculate the Euler characteristic of a 2d orbifold coming from a wallpaper group, we always get zero. And, there are just 17 possibilities!
In fact, wallpaper groups are secretly the same as 2d orbifolds with vanishing Euler characteristic! So, they're not just mathematical curiosities: they're almost as fundamental as 2d manifolds.
The torus and the cylinder, which we've already seen, are two examples. These are well-known to have Euler characteristic zero. Of course, we should be careful: now we're dealing with the cylinder as an orbifold, so the points on the boundary count as "half points" - but its Euler characteristic still vanishes. A more interesting example is the square we get from the group pmm. Let's chop it into vertices, edges, and one face, and figure out how much each of these count:
1/2 1/4-------1/4 | | | | 1/2| 1 |1/2 | | | | 1/4-------1/4 1/2So, the Euler characteristic of this orbifold is
(1/4 + 1/4 + 1/4 + 1/4) - (1/2 + 1/2 + 1/2 + 1/2) + 1 = 0
This is different than the usual Euler characteristic of a rectangle!
As usual, Conway has figured out a charming way to explain all this:
7) John Conway, Peter Doyle, Jane Gilman and Bill Thurston, Geometry and the Imagination in Minneapolis, available at http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/handouts.html
Especially see the sections near the end entitled "Symmetry and orbifolds", "Names for features of symmetrical pattern", "Names for symmetry groups and orbifolds", "The orbifold shop" and "The Euler characteristic of an orbifold". He has a way of naming 2d orbifolds that lets you easily see how they "cost". Any orbifold that costs 2 dollars corresponds to a wallpaper group, and if you list them, you see there are exactly 17!
(I only know two places where the number 17 played an important role in mathematics — the other is much more famous.)
Why exactly 2 dollars? This is related to this formula for the Euler characteristic of a g-handled torus:
V - E + F = 2 - 2g
where g is the number of handles. At the orbifold shop, each handle costs 2 dollars. So, if you buy one handle, you're done: you get an ordinary torus, which has Euler characteristic zero. This is the result of taking the plane and modding out by the most boring wallpaper group in the world. If you don't waste all your money on a handle, you can buy more interesting orbifolds.
Devoted readers of This Week's Finds can guess why I'm talking about this. It's not just that I like the Alhambra. The usual Euler characteristic is a generalization of the cardinality of a finite set that allows negative values - but not fractional ones. There's also something called the "homotopy cardinality" of a space, which allows fractional values - but not negative ones!
If we combine these ideas, we get the orbifold Euler characteristic, which allow both negative and fractional values. This has various further generalizations, like the Euler characteristic of a differentiable stack - and Leinster's Euler characteristic of a category, explained in "week244". We should be able to use these to categorify a lot of math involving rational numbers.
But, it's especially cool how this game of listing all 2d orbifolds with Euler characteristic 0 fits together with things like the "Egyptian fractions" approach to ADE Dynkin diagrams - as explained in "week182". Here I used the Euler characteristic to list all ways to regularly tile a sphere with regular polygons. This gave the McKay correspondence linking Platonic solids to the simply-laced Lie algebras An, Dn, E6, E7 and E8. Taking different values of the Euler characteristic, the same idea let us classify regular tilings of the plane or hyperbolic plane by regular polygons, and see how these correspond to "affine" or "hyperbolic" simply-laced Kac-Moody algebras. Even better, compact quotients of these tilings give some very nice modular curves, like Klein's quartic curve:
8) John Baez, Klein's quartic curve, http://math.ucr.edu/home/baez/klein.html
So, there a lot of connections to be made here... and I can tell I haven't made them all yet! Why don't you give it a try?
To add to the fun, my friend Eugene Lerman has just written a very nice survey paper on orbifolds:
9) Eugene Lerman, Orbifolds as stacks?, available as arXiv:0806.4160.
This describes some deeper ways to think about orbifolds. For example, when we form an orbifold by taking the plane and modding out by a wallpaper group, we shouldn't really say two points on the plane become equal if there's a symmetry carrying one to the other. Instead, we should say they are isomorphic - with the symmetry being the isomorphism. This gives us a groupoid, whose objects are points on the plane and whose morphisms are symmetries taking one point to another. It's a "Lie groupoid", since there's a manifold of objects and a manifold of morphisms, and everything in sight is smooth.
So, orbifolds can be thought of as Lie groupoids. This leads to the real point of Lerman's paper: orbifolds form a 2-category! This should be easy to believe, since there's a 2-category with
In "week75" and "week80" I explained the closely related 2-category with categories as objects; the same idea works for groupoids. So, to get a 2-category of Lie groupoids, we just need to take this idea and make everything "smooth" in a suitable sense.
This turns out to be trickier than you might at first think - and that's where "differentiable stacks" come in. I should explain these someday, but not today. For now, try these nice introductions:
10) J. Heinloth, Some notes on differentiable stacks, Mathematisches Institut Universitität Göttingen, Seminars 2004-2005, ed. Yuri Tschinkel, 1-32. Available as http://www.math.nyu.edu/~tschinke/WS04/pdf/book.pdf or separately as http://www.uni-essen.de/~hm0002/stacks.pdf
12) Kai Behrend and Ping Xu, Differentiable stacks and gerbes, available as arXiv:math/0605694.
Now, besides the Alhambra, Granada also has a wonderful Department of Algebra. Yes - a whole department, just for algebra! And this department has many experts on categorical groups, also known as 2-groups. So it's worth noting that there are 2-groups lurking in the Alhambra.
Any object in any category has a group of symmetries. Similarly, any object in any 2-category has a 2-group of symmetries. So, any orbifold has a 2-group of symmetries. We should be able to get some interesting 2-groups this way.
The group of all symmetries of a manifold - its "diffeomorphism group" - is quite huge. That's because you can warp it and bend it any way you like, as long as that way is smooth. Similarly, the 2-group of all symmetries of an orbifold will often be quite huge.
To cut down the symmetry group of a manifold, we can equip it with a Riemannian metric - a nice distance function - and then consider only symmetries that preserve distances. We can get lots of nice groups this way, called "isometry groups". For example, the group E8, which has been in the news disturbingly often of late, is the isometry group of a 128-dimensional Riemannian manifold called the "octooctonionic projective plane".
So, maybe we can get some nice 2-groups as "isometry 2-groups" of "Riemannian orbifolds". Of course, for this to make sense, we need to know what we mean by a Riemannian metric on on orbifold! I'm no expert on this, but I'm pretty sure the idea makes sense. And I'm pretty sure that the 2d orbifold we get from a specific wallpaper pattern has a Riemannian metric coming from the usual distance function on the plane.
(Warning: the same wallpaper group can arise as symmetries of wallpaper patterns that are different enough to give different Riemannian orbifolds!)
So, here's a potentially fun question: what 2-groups show up as isometry 2-groups of Riemannian orbifolds coming from wallpaper patterns? Try to work out some examples. I don't expect the answer to be staggeringly profound - but it sets up a link between the Alhambra and 2-groups, and that's cool enough for me!
By the way, I obtained copies of some very interesting theses in Granada:
13) Antonio Martinez Cegarra, Cohomologia Varietal, Ph.D. thesis, Departamento de Algebra y Fundamentos, Universidad de Santiago de Compostela.
14) Pilar Carrasco, Complejos Hipercruzados: Cohomologia y Extensiones, Ph. D. thesis, Cuadernos de Algebra 6, Departamento de Algebra y Fundamentos, Universidad de Granada, 1987.
Antonio Cegarra was the one who brought 2-group theory to Granada, and Pilar Carrasco was his student. It's unfortunate that these theses come from the day before electronic typesetting. Luckily, Carrasco's was later turned into a paper:
15) Pilar Carrasco and Antonio Martinez Cegarra, Group-theoretic algebraic models for homotopy types, Jour. Pure Appl. Algebra 75 (1991), 195-235. Also available at http://www.sciencedirect.com/science/article/pii/002240499190133M (click "Download PDF" at top left).
This tackles the ever-fascinating, ever-elusive problem of taking the information in the homotopy type of a topological space and packaging it in some manageable way. If our space is connected, with a chosen basepoint, and it has vanishing homotopy groups above the 2nd, a 2-group will do the job quite nicely! The same idea should work for numbers larger than 2, but n-groups get more and more elaborate as n increases. Carrasco and Cegarra package all the information into a "hypercrossed complex", and I would really like to understand this better.
Carrasco and Cegarra's paper is quite dense. So, I'm very happy to hear that Carrasco plans to translate her thesis into English!
Before I finish, let me mention one more paper about 2-groups:
16) João Faria Martins, The fundamental crossed module of the complement of a knotted surface, available as arxiv:0801.3921.
Martins was unable to attend the Barcelona workshop on 2-groups, but I met him later in Lisbon, and he explained some of the ideas here to me.
A crossed module is just another way of thinking about a 2-group. So, translating the language a bit, the basic concept behind this paper is the "fundamental 2-group" of (X,A,x). Here X is a topological space that contains a subspace A that contains a point x.
Here's how it goes. A 2-group is a 2-category with one object:
xa bunch of morphisms:
f x-------->x(which must all be invertible), and a bunch of 2-morphisms:
f ---->--- / \ x T x \ / ---->---- g(which must also be invertible).
So, we get the fundamental 2-group of (X,A,x) as follows:
If you let A be all of X, you get the fundamental 2-group of (X,x), and this is what people mean when they say connected pointed homotopy 2-types are classified by 2-groups. But the generalization is also quite nice.
In the above paper, Martins uses this generalization, and a bunch of other ideas, to give an explicit presentation of the fundamental 2-group of the complement of a 2-knot (a sphere embedded in 4d Euclidean space). In a certain sense this generalizes the usual "Wirtinger presentation" of the fundamental group of the complement of a knot. But, it's a bit different.
Addenda: This looks like a good book for fans of Islamic tile patterns:
17) Eric Broug, Islamic Geometric Patterns, Thames & Hudson, 2008.
Also, James Propp suggests that people interested in tilings and orbifolds and the work of Conway may enjoy this book:
18) John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss, The Symmetries of Things, A. K Peters Limited, 2008.
I should mention the definition of a wallpaper group: it's just a discrete subgroup of the isometry group of the plane that includes translations in two linearly independent directions. We need the right equivalence relation on these groups to get just 17 of them: they're equivalent if you can conjugate one by an affine transformation of the plane and get the other.
Alas, over at the n-Category Café Richard Hepworth has shown that all the isometry 2-groups of orbifolds coming from wallpaper groups are equivalent to mere groups. It's a pity! But, at least his remarks shed a lot of light on the general theory of isometry 2-groups. First he wrote:
Here is a recipe for computing the isometry 2-groups of a Riemannian orbifold X//G, with G a discrete group acting on a connected simply connected manifold X. I think these are precisely the sorts of orbifolds you are interested in. Apologies for the nasty presentation!Then he wrote:
There is one object.
The arrows are pairs (f,φ) where f is an isometry of X, φ an automorphism of G, and f is equivariant with respect to φ, i.e. f(gx) = φ(g)f(x).
The 2-arrows from (f,φ) to (k,κ) are the elements g of G for which k = gf and κ = g φ g-1.
I'll leave you to work out the various composition maps.
Of course, I haven't told you what a Riemannian metric on an orbifold (or groupoid or stack) is. You can find definitions in various places, but in the case in point these things are all equivalent to putting a G-invariant metric on X. (Does one always exist? Yes, but the fact that G acts with finite stabilizers is essential here.)
I also haven't said how I get the above answer. If you already knew the answer for X = point then the above would be your first guess. The reason the guess is correct is thanks to the assumption that X is connected and simply connected: all G-valued functions on X are constant, and all G-bundles are trivial.
Maybe if you're interested I can go into more detail.
Hmm, John and David's discussion has reminded me of a result I once proved:Actually not all the orbifolds coming from wallpaper groups are "effective" in the above sense. For the orbifold to be effective, I believe the corresponding group must include eitherreflections or glide reflections across two different axes, or a rotation by less than 180 degrees. For example, the torus and "cylinder" described above are noneffective orbifolds. But, most of the interesting examples are covered by Hepworth's results, and the others seem also to have isometry 2-groups that are equivalent to mere groups.
Suppose we are given a morphism f:X → Y of orbifolds and a 2-automorphism φ:f => f. Then φ is trivial if and only if its restriction to any point of X is trivial.
One consequence of this is the following:
Let X be an "effective" orbifold. Then any self-equivalence of X has no nontrivial automorphisms.
And in particular:
The isometry 2-group of an effective orbifold is equivalent to a group.
What does "effective" mean? It means that the automorphism group of any point in X acts effectively on its tangent space, and consequently that almost all points of the orbifold have no inertia at all. This includes all of the orbifolds you are discussing. The fact that the isometry 2-groups are all equivalent to groups is apparent in the description I gave earlier: if we have a 2-arrow g:(f,φ) => (f,φ) then gf = f, so that g is the identity.
Of course, there are many interesting noneffective orbifolds. Some of them are called gerbes. (Gerbes with band a finite group.) Maybe you want to compute their isometry 2-groups? Here's a fact for you:
The isometry 2-group of the nontrivial Z/2-gerbe over S2 is itself a Z/2-gerbe over O(3), the isometry group of S2. In particular, it is not equivalent to any group.
The gerbe over O(3), however, is trivial. But is there a trivialization that respects the 2-group structure?
For details try:
19) Richard Hepworth, The age grading and the Chen-Ruan cup product, available as arXiv:0706.4326.
Richard Hepworth, Morse inequalities for orbifold cohomology, available as arXiv:0712.2432.
According to Hepworth, the first paper "contains the little fact that implies that the automorphism 2-group of an effective orbifold is equivalent to a group". Of course it contains lots of other stuff, too! The second discusses Morse theory on differentiable Deligne-Mumford stacks (these are the proper étale ones). It defines Morse functions, vector fields and Riemannian metrics on differentiable DM stacks. It also proves that Morse functions are generic and that vector fields can be integrated.
You can see more discussion of this Week's Finds at the n-Category Café.
© 2008 John Baez