- #1
John Baez
Also available at http://math.ucr.edu/home/baez/week267.html
July 23, 2008
This Week's Finds in Mathematical Physics (Week 267)
John Baez
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
Some people claim 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 learn 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 a popular zig-zag method of
laying bricks, shown in the article above. The only rotations in this
group are 180-degree rotations: you can rotate any brick 180 degrees
around its center, and the pattern comes back looking the same. There
are no reflections in this group. But, there are "glide reflections"
in two diagonal directions: a "glide reflection" is a combination of a
translation along some line and a reflection across that line.
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. This group contains no reflections
and no rotations. It contains translations along one axis and "glide
reflections" along another.
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 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 2pi but 2pi/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!
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 let's 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 a
spectacularly dull wallpaper group. 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 ways
to regularly tile a sphere with regular polygons. This gave the McKay
correspondence linking Platonic solids to the simply-laced Lie
algebras A_n, D_n, E_6, E_7 and E_8. 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
groupoids as objects,
functors as morphisms, and
natural transformations as 2-morphisms.
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 Ceggara, 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. theis, 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 Ceggara, Group-theoretic
algebraic models for homotopy types, Jour. Pure Appl. Algebra 75
(1991), 195-235. Also available as
http://www.ugr.es/~acegarra/Group theoretic algebraic models for homotopy types.pdf
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,*). Here X is
a topological space that contains a subspace A that contains a point *.
Here's how it goes. A 2-group is a 2-category with one object:
*
a bunch of morphisms:
f
*-------->*
(which must all be invertible), and a bunch of 2-morphisms:
f
---->---
/ \
* T *
\ /
---->----
g
(which must also be invertible).
So, we get the fundamental 2-group of (X,A,*) as follows:
Let the only object be the point *.
Let the morphisms be paths in A starting and ending at *.
Let the 2-morphisms be homotopy classes of paths of paths in X.
If you let A be all of X, you get the fundamental 2-group of (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: I should mention the definition of a wallpaper group: it's 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 Cafe 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!
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, i.e.
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.
Then he wrote:
Hmm, John and David's discussion has reminded me of a result I once
proved:
Suppose we are given a morphism f:X -> Y of orbifolds and a
2-automorphism phi:f => 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) => (f,phi) 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 S^2 is itself
a 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?
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.
For details try:
17) 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 etale 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
Cafe:
http://golem.ph.utexas.edu/category/2008/07/this_weeks_finds_in_mathematic_28.html
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
July 23, 2008
This Week's Finds in Mathematical Physics (Week 267)
John Baez
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
Some people claim 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 learn 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 a popular zig-zag method of
laying bricks, shown in the article above. The only rotations in this
group are 180-degree rotations: you can rotate any brick 180 degrees
around its center, and the pattern comes back looking the same. There
are no reflections in this group. But, there are "glide reflections"
in two diagonal directions: a "glide reflection" is a combination of a
translation along some line and a reflection across that line.
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. This group contains no reflections
and no rotations. It contains translations along one axis and "glide
reflections" along another.
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 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 2pi but 2pi/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!
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 let's 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 a
spectacularly dull wallpaper group. 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 ways
to regularly tile a sphere with regular polygons. This gave the McKay
correspondence linking Platonic solids to the simply-laced Lie
algebras A_n, D_n, E_6, E_7 and E_8. 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
groupoids as objects,
functors as morphisms, and
natural transformations as 2-morphisms.
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 Ceggara, 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. theis, 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 Ceggara, Group-theoretic
algebraic models for homotopy types, Jour. Pure Appl. Algebra 75
(1991), 195-235. Also available as
http://www.ugr.es/~acegarra/Group theoretic algebraic models for homotopy types.pdf
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,*). Here X is
a topological space that contains a subspace A that contains a point *.
Here's how it goes. A 2-group is a 2-category with one object:
*
a bunch of morphisms:
f
*-------->*
(which must all be invertible), and a bunch of 2-morphisms:
f
---->---
/ \
* T *
\ /
---->----
g
(which must also be invertible).
So, we get the fundamental 2-group of (X,A,*) as follows:
Let the only object be the point *.
Let the morphisms be paths in A starting and ending at *.
Let the 2-morphisms be homotopy classes of paths of paths in X.
If you let A be all of X, you get the fundamental 2-group of (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: I should mention the definition of a wallpaper group: it's 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 Cafe 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!
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, i.e.
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.
Then he wrote:
Hmm, John and David's discussion has reminded me of a result I once
proved:
Suppose we are given a morphism f:X -> Y of orbifolds and a
2-automorphism phi:f => 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) => (f,phi) 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 S^2 is itself
a 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?
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.
For details try:
17) 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 etale 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
Cafe:
http://golem.ph.utexas.edu/category/2008/07/this_weeks_finds_in_mathematic_28.html
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