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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 [Broken]

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 lets you easily see how they "cost".

Any orbifold that costs 2 dollars corresponds to a wallpaper

group, and if you list them, you see there are exactly 17!

(I only know two places where the number 17 played an important

role in mathematics - the other is much more famous.)

Why exactly 2 dollars? This is related to this formula for the Euler

characteristic of a g-handled torus:

V - E + F = 2 - 2g

where g is the number of handles. At the orbifold shop, each

handle costs 2 dollars. So, if you buy one handle, you're done:

you get an ordinary torus, which has Euler characteristic zero.

This is the result of taking the plane and modding out by 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 [Broken] 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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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

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

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