## July 23, 2008

### This Week’s Finds in Mathematical Physics (Week 267)

#### Posted by John Baez

In week267 of This Week’s Finds see the tile patterns of the Alhambra:

Then learn about the 17 wallpaper groups, their corresponding 2d orbifolds, the role of 2-groups as symmetries of orbifolds, the work of Carrasco and Cegarra on hypercrossed complexes, and the work of João Faria Martins on the fundamental 2-group of a 2-knot.

Posted at July 23, 2008 3:31 PM UTC

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### Re: This Week’s Finds in Mathematical Physics (Week 267)

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.

So they get to be called Euclidean 2-orbifolds.

Hmm, nice looking notes. Looks like you need to put Morse theory into the mix:

We sincerely believe that critical-net and critical-net-on-orbifold crystallographic illustrations will some day be of fundamental importance in crystal chemistry and crystallographic topology. Critical nets are based on the concepts of Morse functions, Morse theory (i.e., critical point analysis), and hyperplane arrangements in topological complexes, which are classic topics in the mathematical topology and global analysis literature. Morse functions on orbifolds constitute a relatively new aspect of equivariant (i.e., group orbit compatible) topology.

Posted by: David Corfield on July 23, 2008 4:29 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

I only know two places where the number 17 played an important role in mathematics – the other is much more famous.

The Feller number?

Posted by: John Armstrong on July 23, 2008 8:03 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

According to Wikipedia:

17 is known as the Feller number, after the famous mathematician William Feller who taught at Princeton University for many years. Feller would say, when discussing an unsolved mathematical problem, that if it could be proved for the case n = 17 then it could be proved for all positive integers n. He would also say in lectures, “Let’s try this for an arbitrary value of n, say n=17.”

The fact John may have been hinting at would then lead you to believe that all regular polygons could be constructed by ruler and compass.

Posted by: David Corfield on July 24, 2008 11:51 AM | Permalink | Reply to this

### Janice Ian: I learned the truth at seventeen; Re: This Week’s Finds in Mathematical Physics (Week 267)

17: at Prime Curios which begins:

The 17-stringed bass koto (or jūshichi-gen, literally “seventeen strings”) is used in the album PRIME NUMBERS by Neptune/Watanabe. As a matter of fact, all the traditional Japanese instruments played in it contain prime numbers. The other instruments used are the 3-stringed shamisen, the 5-holed shakuhachi (bamboo flute), and the 13-stringed koto.

Weisstein, Eric W. “Constructible Polygon.” From MathWorld–A Wolfram Web Resource.

Posted by: Jonathan Vos Post on July 24, 2008 5:39 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Of course I was hinting at Gauss’ proof that regular polygons with $2^{2^n} + 1$ sides can be constructed with ruler and compass. The construction of the regular pentagon goes back at least to Euclid, but some say Gauss wanted a regular heptadecagon to be inscribed on his grave.

Is it a coincidence that Johannes Erchinger’s construction takes 64 steps, a power of two?

Posted by: John Baez on July 25, 2008 11:18 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Of course I was hinting at Gauss’ proof that regular polygons with $2^{2^{n+1}}$ sides can be constructed with ruler and compass.

It’s not terribly important, but I don’t think that’s quite right. I think the actual point is that a for a prime $p$, a $p^{th}$ root of unity is constructible (in the ruler and compass sense) if and only if $p-1$ (the order of the Galois group) is a power of 2, which in turn means it must be of the form $2^{2^n}$, in which case $p$ is called a Fermat prime. An $n^{th}$ root of unity is constructible iff it is a product of a power of 2 and distinct Fermat primes, if I recall correctly.

For instance, $2^32 + 1$ is not prime, and a polygon with that many sides is not constructible.

Posted by: Todd Trimble on July 25, 2008 7:13 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

At the orbifold shop, each handle costs 2 collars.

I know geometric topologists talk about collars, but presumably this should be dollars.

Posted by: David Corfield on July 24, 2008 11:28 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

A collar, a dollar, a slovenly scholar…

Fixed! Thanks!

Posted by: John Baez on July 25, 2008 11:38 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Yesterday the Proofs, Programs and Systems group had a little microconference:

Bonjour,

Nous profitons de la visite de John Baez à notre équipe PPS pour organiser à Chevaleret une journée “n-catégories, n-groupoïdes et logique”.

La journée se tiendra jeudi 24 juillet, en salle 6C01.

Le programme de la journée est le suivant:

• 10h – Mark Weber : A tutorial on monads with arities
• 11h – Nicolas Tabareau : Computing free models as Kan extensions
• 14h – John Baez : Groupoidification
• 15h – Dimitri Ara : Weak infinity-categories : Grothendieck versus Batanin
• 16h30 – Samuel Mimram : A tutorial on polygraphs and their applications to semantics
• 17h30 – Jonas Frey : a 2-dimensional adjunction between triposes and toposes

En espérant vous voir nombreux jeudi,

Paul-André Melliès

Jamie Vicary came all the way from London to attend, getting up at a frightfully early hour to catch the train to Paris.

It was a lot of fun! Dimitri Ara’s talk was of special historical interest, because his advisor Maltsionitis has pored through Grothendieck’s Pursuing Stacks, extracted a definition of weak $\infty$-groupoids, generalized it to weak $\infty$-categories… and now Ara has compared it with Batanin’s definition, finding it to be equivalent, but phrased in a wholly different language: the language of sketches!

But, I admit that at moments when my attention flagged, I started working out the 2d orbifolds coming from various wallpaper groups. This made me realize how fun the subject actually is… so today I put a bunch more material about this in This Week’s Finds. I’ll post it as a comment below, for those who’ve already read the earlier version.

By the way, my request to compute the symmetry 2-groups of some of the resulting Euclidean 2-orbifolds was mainly directed at David Corfield (since it ties into our Klein 2-Geometry thread) and Eugene Lerman (since it should be easy for him). Maybe some partially worked-out examples will help… though I didn’t do the most interesting examples. In fact, I’m not sure I see any interesting 2-groups! I’m a bit confused, but I only see the 1-groups.

Posted by: John Baez on July 25, 2008 12:05 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Here’s some stuff I added to week267, which should make 2d orbifolds easier to understand:

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     R


This 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     R


Similarly, 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     R


So, 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     T


The 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     T


and 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     T


but 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  T


There 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  .
TTTT


So, 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:

• We count edges and vertices on the boundary for 1/2 the usual amount.
• We count a cone point of order n as 1/nth of a point.
• We count a corner reflector of order 2n as 1/(2n)th of a point.

The idea is that these features have been “folded over” by a certain amount, so they count for a fraction of what they otherwise would.

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
1/2

So, 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!

Posted by: John Baez on July 25, 2008 12:47 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

In that orbifold you’ve just mentioned, aren’t there some 2-morphisms to be had from, say, the identity 1-morphism to itself?

If we were just looking at a vertex with its four arrows, then AUT of it would have some stuff going on at the 2-level. Is there a problem in the full orbifold with continuous assignments of natural transformations at the different points?

Eugene suggested a ‘homogeneous’ orbifold here, which we might need for ‘transitivity’.

Posted by: David Corfield on July 25, 2008 12:55 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

I urge people, especially David and Eugene, to compute the 2-group of symmetries of the above Riemannian orbifold: the rectangle with $\mathbb{Z}/2$ symmetries along edges and $\mathbb{Z}/2 \times \mathbb{Z}/2$ symmetries at the the corners.

By the way: I like how the two $\mathbb{Z}/2$ symmetries along the vertical and horizontal edges, corresponding to reflections across those edges, merge to form the $\mathbb{Z}/2 \times \mathbb{Z}/2$ symmetry group at the corners. For one thing, this helps me understand something that previously seemed like mysterious abstract nonsense: the complex of groups associated to an orbifold.

Posted by: John Baez on July 25, 2008 12:57 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Wow, David started computing the 2-group of symmetries of this Riemannian orbifold:

          1/2
1/4-------1/4
|         |
|         |
1/2|    1    |1/2
|         |
|         |
1/4-------1/4
1/2


before I even officially urged him to do it!

Is there a problem in the full orbifold with continuous assignments of natural transformations at the different points?

That’s exactly the issue that has me confused!

There are various subtleties. At first, I wanted to ask: how are we making the above gadget into a topological (or smooth) groupoid? What’s the right topology (or smooth structure) on the space of morphisms? Is it some sort of quotient topology?

Since the four endomorphisms at the corners collapse instantaneously to one as we move into the body of the rectangle, this topology would seem to be non-Hausdorff. After all, a classic example of a non-Hausdorff space is the line with two points at the origin, sharing all the same neighborhoods.

But in fact, this question of the correct topology might almost be a red herring! After all, we should expect to need the stacky perspective to settle issues like these. And as a stack, the orbifold above is equivalent to the weak quotient $\mathbb{R}^2//\Gamma$, where $\Gamma =$ pmm is the relevant wallpaper group. So, we should be able to answer my question using $\mathbb{R}^2//\Gamma$, which doesn’t have any weird non-Hausdorff stuff going on.

Posted by: John Baez on July 25, 2008 1:13 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

I seem to be seeing only 1-groups, like you did.

Perhaps that’s why back here, I cut the vertices adrift.

Posted by: David Corfield on July 25, 2008 2:47 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

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!

There is one object.

The arrows are pairs (f,phi) where f is an isometry of X, phi an automorphism of G, and f is equivariant with respect to phi, ie f(gx)=phi(g)f(x).

The 2-arrows from (f,phi) to (k,kappa) are the elements g of G for which k=gf and kappa=g phi 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.

Posted by: Richard Hepworth on July 25, 2008 3:33 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Let’s just look at that in itex:

The arrows are pairs $(f, \phi)$ where $f$ is an isometry of $X$, $\phi$ an automorphism of $G$, and $f$ is equivariant with respect to $\phi$, ie $f(g x) = \phi(g) f(x)$.

The 2-arrows from $(f, \phi)$ to $(k, \kappa)$ are the elements $g$ of $G$ for which $k = g f$ and $\kappa = g \phi g^{-1}$.

Posted by: David Corfield on July 25, 2008 4:58 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Note that this is a Lie 2-group. John, do you know what its 2 Lie algebra looks like?
Posted by: Eugene Lerman on July 25, 2008 7:30 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Eugene wrote:

Note that this is a Lie 2-group. John, do you know what its 2 Lie algebra looks like?

Hi! A strict Lie 2-group has a Lie group of arrows and a Lie group of morphisms 2-arrows, and we get its Lie 2-algebra by taking the Lie algebras of both these things.

So, I’m afraid the Lie 2-algebra of the Lie 2-group Richard described is rather dull — always equivalent to a Lie algebra — since so far we’re assuming $G$ is a discrete group.

Namely: let $Isom(X)$ be the Lie group of isometries of $X$ and $isom(X)$ its Lie algebra, whose elements are called ‘Killing vector fields’. Let $G$ be a Lie group acting as isometries of $X$. This gives an action of $G$ on $Isom(X)$ and thence $isom(X)$.

When $G$ is discrete, I bet the Lie 2-algebra of isometries of the orbifold $X//G$ is equivalent to $isom(X)$.

I guessed this by taking Lie algebras of all the groups in the 2-group Richard described.

What happens when $G$ is itself a Lie group? Is $X//G$ sometimes or always an orbifold… or some more general Lie groupoid whose isometry 2-group can still be computed by Richard’s prescription?

Posted by: John Baez on July 27, 2008 12:14 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

John wrote:

What happens when G is itself a Lie group? Is X//G sometimes or always an orbifold… or some more general Lie groupoid whose isometry 2-group can still be computed by Richard’s prescription?

If $G$ acts with finite stabilizers then the quotient is an orbifold; if it acts with discrete stabilizers then the quotient is still an etale stack; if we know nothing more than that $G$ is a compact Lie group, then the result is a proper differentiable stack’.

In general, even in the first case, you can’t compute the isometries from my prescription; I had to use connectivity and simple connectivity of $X$. And no, I haven’t computed an example where this fails!

For the other cases, be very careful, for the following reason. Orbifolds have tangent bundles and those tangent bundles always admit a metric. So you can talk about isometries. Etale stacks do have tangent bundles, but these don’t necessarily admit metrics. So you can’t talk about isometries unless you know you’ve got a metric. General proper stacks are worse: they have tangent stacks, but these aren’t necessarily vector bundles.

I am, however, working on this last aspect. I hope to elaborate on this in a reply to your other comment.

Posted by: richard hepworth on July 28, 2008 5:34 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

John’s use of T and R etc reminded me of an alternate name for frieze patterns which may not have the intellectual content of pmm or whatever but are much easier to remember what they correspond to. i learned them from Tom Brylawski at UNC-CH, whose passing I deeply regret.

For maximal symmetry, the label is O for OOOO

For translation and up-down symmetry C for CCCC

etc

Posted by: jim stasheff on July 26, 2008 1:08 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Hi!

I have been praying for years now that John Baez writes a book where all ( or at least some) of this marvelous stuff from his twf columns is tied together.

But I’m afraid this day won’t come.

Posted by: Luis Herzsprung on July 25, 2008 3:27 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Luis wrote:

I have been praying for years now that John Baez writes a book where all ( or at least some) of this marvelous stuff from his twf columns is tied together.

Thanks! I’ve been praying this too. Maybe if enough of us join in, I’ll be allowed to do this in the afterlife, when I’ll have more time.

Actually, I’m afraid that even if I lived forever, I’d still be way too busy to do everything I want to do, unless I could temporarily alter my personality with the help of an ‘exoself’, as Greg Egan describes in Permutation City. He has a character who spends a year making tables, then a year collecting insects… Right now I’d like to spend a year studying 2d orbifolds. But next week I’ll be onto something else.

Posted by: John Baez on July 25, 2008 3:49 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Hmm, John and David’s discussion has reminded me of a result I once proved:

Suppose we are given a morphism $f: X\to Y$ of orbifolds and a 2-automorphism $\phi: f\Rightarrow f$. Then $\phi$ 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 $g: (f,\phi)\Rightarrow(f,\phi)$ then $g f = f$, 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 $\mathbb{Z}_2$-gerbe over $S^2$ is itself a $\mathbb{Z}_2$-gerbe over $O(3)$, the isometry group of $S^2$. 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?

Posted by: richard hepworth on July 25, 2008 8:23 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Your last example sounds similar to what I was trying to work out at the end of this post.

Is that right to think that for sub-2-groups of your isometry 2-group we can have, say, both $O(2)$ and a $\mathbb{Z}_2$-gerbe over $O(2)$?

Posted by: David Corfield on July 26, 2008 2:54 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Is that right to think that for sub-2-groups of your isometry 2-group we can have, say, both O(2) and a $\Z_2$-gerbe over O(2)?

First I need to correct myself:

Let $\mathbb{G}$ denote the nontrivial $\mathbb{Z}_2$-gerbe on $S^2$, equipped with the metric inherited from the usual metric on $S^2$. The isometry 2-group of $\mathbb{G}$ is equipped with a morphism to the isometry group of $S^2$: $\mathrm{Isom}(\mathbb{G})\to O(3).$ Then $\mathrm{Isom}(\mathbb{G})$ is the $\mathbb{Z}_2$-gerbe over $O(3)$ that is nontrivial on each component.

David, the answer to your question is yes. The part of the 2-group that lies over $O(2)$ is a $\mathbb{Z}_2$-gerbe; but this gerbe is trivial, so has a section, so that $O(2)$ is itself a subgroup.

Thanks to Eugene for requesting details of this computation. He’s getting them by email, but I’d be happy to supply them to anyone else.

Posted by: richard hepworth on July 29, 2008 12:02 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Thanks! Next step some quotienting of 2-groups by sub-2-groups. Let’s see if I can lure you into the Klein 2-geometry project.

Posted by: David Corfield on July 29, 2008 12:46 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Richard wrote:

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.

Thanks for crushing my attempt to get interesting 2-groups from wallpaper groups by showing these 2-groups are all equivalent to mere groups!

It’s a bit sad, but better to be sad and wise than entertain false hopes.

By the way, I don’t think all the orbifolds coming from wallpaper groups are “effective” in the above sense. I believe one of these orbifolds is effective iff the corresponding wallpaper group includes either reflections and/or glide reflections across two different axes, or a rotation by less than 180 degrees. But, the remaining examples are so dull that I bet all their isometry 2-groups are mere groups as well.

For example, I think the torus $T^2$ and “cylinder” $T^2//(\mathbb{Z}/2)$ that I described are noneffective orbifolds. But, I think their isometry 2-groups are still equivalent to groups — that’s obvious in the first case and probably still true in the second.

Are any of the results you mention written up and publicly available? It would be nice to advertise them here, for people who wish to ponder these issues further.

Posted by: John Baez on July 27, 2008 11:50 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Are any of the results you mention written up and publicly available? It would be nice to advertise them here, for people who wish to ponder these issues further.

I have two articles on the subject, both available on my homepage.

The first, “The age grading and the Chen-Ruan cup product”, contains the little fact that implies that the automorphism 2-group of an effective orbifold is equivalent to a group.

The second, “Morse Inequalities for Orbifold Cohomology”, contains a full treatment of Morse theory on differentiable Deligne-Mumford stacks. (These are the proper etale ones.) It contains (the first, I think) definitions of 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.

Posted by: richard hepworth on July 29, 2008 12:55 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Thanks for these references, Richard. Nice to see more ‘higher mathematics’ coming out of Sheffield!

(‘Higher mathematics’ is Alan Weinstein’s term for math that uses categorified notions like gerbes, stacks, $n$-categories etc.)

I took a peek at your thesis, which made mew wish I understood a tiny bit more about 3-Sasakian manifolds. Just a tiny bit. The first two clauses in the definition seem to say it’s a Riemannian manifold equipped with three orthonormal vector fields that generate a ‘local action’ of $\mathbb{SO}(3)$. The third clause… well, I have no sense for that.

(By ‘local action’ I mean very roughly that given any point $x$ and a sufficiently small element $x \in \mathbb{SO}(3)$, we get a new point $g x$, and the usual axioms for an action hold when both sides are well-defined.)

Posted by: John Baez on July 29, 2008 1:17 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Has anything been written about gerbe symmetry in general?

Posted by: David Corfield on July 28, 2008 12:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

There is a large literature on equivariant gerbes, meaning gerbes with Lie group actions. Little, as far as a I know, on treating gerbes as objects with 2-groups of automorphisms. After a cursory serach I found

Automorphism 2-group of a weighted projective stack
by Behrang Noohi.

Note that gerbes come in many guises: as bundle gerbes, as groupoid extensions, as certain kinds of stacks…

Posted by: Eugene Lerman on July 28, 2008 6:20 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

In the main twf article, you say

If you let $A$ be all of $X$, you get the fundamental 2-group of $(X,x)$

which unfortunately should read more like “If you let $A = \{x\}$ …”. Compare to the situation of the relative fundamental thingie (=pointed set) $\pi_1(X,A,x)$ Here $x\in A$.

Posted by: David Roberts on July 29, 2008 12:28 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

Dear David and John,

A small correction to the entry in t.w.f.:

In the context of the article which I wrote, what we get from a pair $(X,A,*)$ of pointed spaces is a strict 2-group with a single object, say $P=P(X,A,*)$, so everything in twf should be taken up to homotopy.

Therefore $*$ is the only object of $P$, the morphisms are paths in $(A,*)$ up to homotopy (i.e. elements of $\pi_1(A,*)$), and the 2-morphisms are maps $f: D^2=[0,1]^2\to X$ such that $f(\{0,1\}\times I)=*$ and $f(I \times \{0,1\})\subset A$, up to filtered homotopy; see for example:

“On the Connection between the Second Relative Homotopy Groups of Some Related Spaces,” Ronald Brown and Philip J. Higgins, Proceedings of the London Mathematical Society 1978 s3-36(2):193-212

Given a CW-complex $X$, what is natural (in this construction) is to take $A=X^1$, the 1-skeleton of $X$; Since if A=X then all would get trivial for the same reason why $\pi_2(X,X)$ is trivial.

Then $P(X,X^1,*)$ determines the algebraic and topological 2-type of $X$. It is a discrete object and we can present it by generators and relations, therefore it is particularly suitable for explicit calculations.

Of course there is an ambiguitity due to the choice of a cell decomposition of $X$, however, this dependence is minimal as it is explained in The Fundamental Crossed Module of the Complement of a Knotted Surface ; or in On the homotopy type and the fundamental crossed complex of the skeletal filtration of a CW-complex

What appears in the first of this articles is an algorithm to calculate this 2-group for the case of the complement of a knotted surface in $S^4$.

Posted by: João Faria Martins on July 29, 2008 12:24 PM | Permalink | Reply to this

### A Comment on Orbifolds

There’s been a lot of discussion of orbifolds here and in another recent post. I thought I’d take this opportunity to enthuse about the current state of the subject, particularly as a response to John’s comment

I’m not surprised nobody talks about ‘Morse functors’, because not many people seem to think of orbifolds as groupoids — not yet, anyway. They seem to think of orbifolds as manifold-like things where some points have neighborhoods that look, not like $\mathbb{R}^n$, but like a quotient of $\mathbb{R}^n$ by a finite group action. A space of orbits, in other words — hence the term ‘orbifold’.

The more modern viewpoint, which Eugene is pushing, amounts roughly to saying: “Don’t just take the quotient of $\mathbb{R}^n$ by a finite group action; take the weak quotient, and get a groupoid!” This viewpoint tends to force itself upon people, even the unwilling, as soon as they try to figure out the right notion of morphism between orbifolds.

I think that in fact the more modern viewpoint’, where people think of orbifolds as groupoids or stacks, is far better served than John’s comment would have us believe.

To see the state of the art you could look at the programme of the Workshop on Topological and Differentiable Stacks that took place at the CRM in Barcelona earlier this year. But right now I really just want to list a few topics to whet your appetite.

1) String topology has been a very popular topic in topology in recent years; it is an example of an HCFT and also arises in various breeds of Floer theory. It has been extended to stacks by Behrend et al. in their paper String topology for stacks. Without the stacky viewpoint it would have been very difficult to describe the necessary loopspaces to develop the theory.

2) One fantastic example of an orbifold is the the moduli stack $\overline{M}_{g,n}$ of stable nodal curves of genus $g$ and with $n$ marked points. Ebert and Giansiracusa’s paper Pontrjagin-Thom maps and the homology of the moduli stack of stable curves develops Pontrjagin-Thom maps for stacks uses this to concoct large families of torsion classes in the homology of $\overline{M}_{g,n}$.

3) For me, the most interesting thing about orbifolds at the moment is the work on the crepant resolution conjecture. A crepant resolution $Y$ of an (algebraic) orbifold $X$ is a smoothing’ of the orbifold that is somehow minimal’. The crepant resolution conjecture says that we should be able to predict the topology of $Y$ from an appropriate notion of the `stacky topology’ of $X$. See Coates-Ruan’s Quantum Cohomology and Crepant Resolutions: A Conjecture. This is strongly connected to the McKay correspondence.

Posted by: richard hepworth on July 29, 2008 1:44 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 267)

In: The discrete groups of Monstrous Moonshine,
Conway, McKay, and Sebbar:Proc. Amer. Math. Soc. 132, 2233–2240, (2004).

describe the 171 discrete groups fixing the MM functions.
It would be very useful to have the orbifold notation for
these groups.

Posted by: John McKay on August 6, 2008 12:38 PM | Permalink | Reply to this

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