Also available at http://math.ucr.edu/home/baez/week213.html
April 10, 2005
This Week's Finds in Mathematical Physics - Week 213
John Baez
Here's a book I've been reading lately:
1) Kenneth S. Brown, Cohomology of Groups, Graduate Texts in Mathematics
182, Springer, 1982.
I should have read this book a long time ago - but I probably wouldn't
have enjoyed it as much as I do now. All sorts of things I struggled
to learn for years are neatly laid out here. Best of all, he comes
right out and admits from the start that the cohomology of groups is
really a branch of *topology*, instead of hiding this fact like some
people do.
This is something every mathematician should know: you can take any
group and turn it into a space, thus "reducing" group theory to topology.
In particular, if you have any trick for telling spaces apart, like
"cohomology theory", you can apply it to groups as well.
Of course topology is *harder* than group theory in many ways - hence
my quotes around "reducing". Indeed, algebraic topology was invented
as a trick for reducing topology to group theory! But, the bridge
turns out to go both ways, and there's a lot of profitable traffic in
both directions.
Ultimately, as James Dolan likes to point out, it's all about the unity of
mathematics. Topology is about our concept of *space*, while group theory
is about our concept of *symmetry*... but the amazing fact is that they turn
out to be two aspects of the same big thing! Mathematics is a source of
endless surprises, but this is one of the biggest jaw-droppers of all.
The idea goes back at least to Evariste Galois, who noticed that you can
classify the ways a little thing can sit in a bigger thing by keeping track
of what we now call its "Galois group": the group of all symmetries of the
big thing that map the little thing to itself. For example, you can pick
out a point or line in the plane by keeping track of which symmetries of
the plane map this point or line to itself.
However, the idea of using groups to classify how a little thing sits
in a big one was really made explicit in Felix Klein's "Erlangen program",
a plan for reducing *geometry* to group theory.
You may know Klein for his famous one-sided bottle:
A mathematician named Klein
Thought the Moebius strip was divine.
Said he: "If you glue
The edges of two
You'll get a weird bottle like mine!"
Or maybe you know that the symmetry group of a rectangle, including
reflections, is called the "Klein 4-group":
1 a b c
---------
1| 1 a b c
a| a 1 c b
b| b c 1 a
c| c b a 1
He is also known for some other groups called "Kleinian groups", which
act as symmetries of fractal patterns like this:
2) Jos Leys, Kleinian Pages, http://www.josleys.com/creatures42.htm
If you like cool pictures, check out this website! I've linked you to
the page that most closely connects to Kleinian groups, but there are
lots of other more fanciful pictures. And if you get interested in
the math lurking behind these fractals, you've *got* to try this book:
3) David Mumford, Caroline Series, and David Wright, Indra's Pearls:
The Vision of Felix Klein, Cambridge U. Press, Cambridge, 2002.
Mumford is a world-class mathematician, so this book is completely
different from the superficial descriptions of fractals one often sees
in math popularizations - but it's still readable, and it's packed with
beautiful pictures. You can learn a lot about Kleinian groups from this!
The Kleinian groups arose from Klein's studies of complex functions,
which he considered his best work. But, he was also a mathematical
physicist. Among other things, he wrote a four-volume book on tops
with one of the fathers of quantum mechanics, Arnold Sommerfeld:
4) Felix Klein and Arnold Sommerfeld, Ueber die Theorie des Kreisels,
4 vols, 1897-1910. Reprinted by Johnson, New York, 1965. Available at
http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV7354.0001.001
This came after a book he wrote on his own:
5) Felix Klein, The Mathematical Theory of the Top, Scribner's,
New York, 1887.
This may seem like a lot of books about a kid's toy! But, tops
are profoundly related to the rotation group, and the "exactly
solvable" tops discovered by Euler, Lagrange/Poisson, and
Sofia Kowalevskaya are solvable because of their symmetries -
deeply hidden symmetries, in the case of the Kowalevskaya top.
So, one can imagine why Klein liked this subject.
Klein also wrote a book on the icosahedron and the quintic equation:
6) Felix Klein, Lectures on the Icosahedron and the Solution of Equations
of the Fifth Degree, 1888. Reprinted by Dover, New York, 2003.
Galois had already noticed that the number field you get by taking the
rationals and throwing in the roots of a typical quintic:
ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0
has as its symmetry group all the permutations of the 5 roots. Indeed,
he saw that the "unsolvability" of this group, in a technical sense, is
what makes it impossible to solve the quintic by radicals. It must have
been common knowledge that the symmetry group of the icosahedron is the
group of all *even* permutations of 5 things. But, Klein took this much
further! Alas, I've never really understood what he did. Perhaps if
I read this and think hard, I'll understand:
7) Peter Doyle and Curt McMullen, Solving the quintic by iteration,
Acta Math. 163 (1989), 151-180. Available at
http://math.dartmouth.edu/~doyle/docs/icos/icos/icos.html
Anyway, it should be clear by now that Klein was a lover of symmetry.
He was also a bit of a visionary, and his obituary by Grace Chisholm
Young shows that this got him in some trouble:
One of Weierstrass' pupils, still alive, told me that at Berlin
Klein was anathema: it was said that his work was not mathematics
at all, but mere talk. This criticism shows a want of appreciation
of his rare type of mind. It teemed with ideas and brilliant
reflections, but it is true that his work lacks the stern aspects
required by mathematical exactitude. It was in personal contact that
this was corrected, at least in so far as his students were concerned.
His favourite maxim was, "Never be dull".
In a talk he wrote in 1872 when he was made professor at Erlangen
University - a talk he didn't actually give! - Klein outlined what is now
called his "Erlangen program". The idea here is that different kinds of
geometry correspond to different symmetry groups. Taken to the extreme,
this philosophy says that a geometry is just a group! In a given geometry,
a "figure" of any kind - like a point or line - can be detected by the
subgroup of symmetries that map that figure to itself. So, a figure is
just a subgroup!
This program eventually led to a grand theory of groups and geometry based
on "flag manifolds", which I tried to sketch in "week178", "week180", and
"week181".
It's important to realize how similar the Erlangen program is to Galois
theory. Galois had also used group theory to classify how a little thing
can sit in a bigger thing, but in situations where the "things" in question
are commutative algebras - for example, the rational numbers with some roots
of polynomials thrown in.
Now, commutative algebra is like topology, only backwards. Any space has
a commutative algebra consisting of functions on it, and if we're very
clever we can think of any commutative algebra as functions on some space -
though this was only achieved long after Galois, by Alexander Grothendieck.
What do I mean by "backwards"? Well, suppose you have a "covering
space" - a big space sitting over a little one, like a spiral sitting
over the circle. In this situation, any function on the little space
downstairs defines a function on the big one upstairs. So, the algebra
of functions on the little space sits inside the algebra of functions on
the big space.
Notice how it's backwards. Classifying how a little commutative
algebra can sit in a big one amounts to classifying how a big space
can cover a little one! For more details on this analogy, try "week198",
"week201" and especially "week205".
I should warn you: the Galois group has a different name when we apply
it to the classification of covering spaces - we call it the group
of "deck transformations". The idea is pretty simple. Suppose Y is a
covering space of X, like this:
----------------
---------------- Y
|p
v
---------------- X
We've got a function p: Y -> X, and sitting over each point of X are
the same number of points of Y, living on different "sheets" that look
locally just like X. You should imagine the sheets being able to
twist around from place to place, like the edges of a Moebius strip.
Anyway, a "deck transformation" is just a way of mapping Y to itself
that permutes the different points sitting over each point of X.
The theory of this was worked out by Riemann, Poincare, and others.
Poincare showed you could use this idea to turn any connected space X
into a group - its "fundamental group". There are different ways to
define this, but one is to form the most complicated possible
covering space of X that's still connected - its "universal cover".
Then, take the group of deck transformations of this! Following
Galois' philosophy, all the other connected covering spaces of X
correspond to subgroups of this group.
The theory of the fundamental group was just the beginning when
it came to groups and topology. One of many later big steps, back
in the late 1940s, was due to Sammy Eilenberg and Saunders Mac Lane.
They saw how to reverse the "fundamental group" idea and turn any
group back into a space!
More precisely: for any group G, there's a space whose fundamental group
is G and whose higher homotopy groups vanish. It's sometimes called the
"Eilenberg - Mac Lane space" and denoted K(G,1), but sometimes it's called
the "classifying space" and denoted BG. It's pretty easy to build;
I described how back in "week70".
You start with a point:
o
Then you stick on an edge looping from this point to itself for
each element a in G. Unrolled, it looks like this:
O--a->--O
where a is an element of our group. Then, whenever we have ab = c in
our group, we stick on a triangle like this:
O
/ \
a b
/ \
O--c->--O
Then, whenever we have abc = d in our group, we stick on a tetrahedron
like this:
O
/|\
/ | \
/ b \
a | bc
/ _O_ \
/ / \_ \
/ _ab c_ \
/_/ \_\
O-------d->-------O
And so on, forever! For each list of n group elements, we get an
n-dimensional simplex in our Eilenberg-Mac Lane space. The resulting
space knows everything about the group we started with. In particular,
the fundamental group of this space will be the group we started with!
Using this idea, we can do some fiendish things. For example, for each n
we can form a set C_n(G,A) consisting of all functions that eat
n-dimensional simplices in the Eilenberg-Mac Lane space of G and spit
out elements of some abelian group A. There are maps
d: C_n(G,A) -> C_{n+1}(G,A)
reflecting the fact that each (n+1)-simplex has a bunch of n-simplices
as its faces. Since the boundary of a boundary is zero,
d^2 = 0
Guys who live in the kernel of
d: C_n(G,A) -> C_{n+1}(G,A)
are called "n-cocycles", and guys who live in the image of
d: C_{n-1}(G,A) -> C_n(G,A)
are called "n-coboundaries". Since d^2 = 0, every coboundary is
a cocycle, but not always vice versa. So, we can form the group of
cocycles mod coboundaries. This is called the "nth cohomology
group" of G with coefficients in A, and it's denoted
H^n(G,A).
This sounds unmotivated at first, but the nth cohomology group of a
space is really just a clever way of keeping track of n-dimensional
holes in that space. So, what we're doing here is cleverly defining
a way to study "holes" in a GROUP! There are deeper, more conceptual
ways of understanding group cohomology, but this is not bad for starters.
For example, let's take the simplest group that's not *utterly* dull -
the integers mod 2, or Z/2. Here we get
K(Z/2,1) = RP^infinity
where RP^infinity is the space formed by taking an infinite-dimensional
sphere and identifying opposite points. This space has holes of
arbitrarily high dimension, so the cohomology groups of Z/2 go on being
nontrivial for arbitrarily high n. I sketched a "picture proof" here:
8) John Baez, Fall 2004 Quantum Gravity Seminar, week 10, notes
by Derek Wise, http://math.ucr.edu/home/baez/qg-fall2004/
and I showed that, for example
Z if n = 0
H^n(Z/2,Z) = 0 if n is odd
Z/2 if n is even and > 0
I also explained how this stuff is related to topological quantum field
theory.
Anyway, all this is just the very superficial beginnings of the subject
of group cohomology. Read Brown's book to dig deeper!
Personally, what I find most exciting about this book now are the
remarks on the "Euler characteristic" of a group. Let me explain this...
though now I'll have to pull out the stops and assume you know some
group cohomology.
We can try to define the "Euler characteristic" of a group G to be the
Euler characteristic of K(G,1). This is the alternating sum of the
dimensions of the rational cohomology groups
H^n(G,Q)
Of course, this alternating sum only converges if the cohomology
groups vanish for big enough n. Also, they all need to be
finite-dimensional.
Unfortunately, not many groups have well-defined Euler characteristic
with this naive definition!
For example, people have studied groups G whose nth cohomology vanishes
for n > d, regardless of the coefficients. If we take the smallest
d for which this holds, such a group G is said to have "cohomological
dimension" d. Eilenberg and Ganea showed that for d >= 3, a group has
cohomological dimension d whenever we can build K(G,1) as a simplicial
complex (or CW complex) with no cells of dimension more than d.
This is a nice geometrical interpretation of the cohomological dimension.
But, one can show that groups with torsion never have finite cohomological
dimension! We've seen an example already: Z/2, whose Eilenberg-Mac Lane
space is infinite-dimensional.
However, it turns out that there's a generalization of the Euler
characteristic that makes sense for any group G that has a torsion-free
subgroup H whose Euler characteristic is well-defined in the naive way,
as long as H has finite index in G. We just define the Euler characteristic
of G to be the Euler characteristic of H divided by the index of H in G.
Take my favorite example, SL(2,Z). This has torsion, so its cohomological
dimension is infinite and its naive Euler characteristic is undefined!
Indeed, I wrote a whole issue of This Week's Finds about some elements of
orders 4 and 6 sitting inside SL(2,Z), related to the symmetries of square
and hexagonal lattices - see "week125".
But, SL(2,Z) has a torsion-free subgroup of index 12, namely its
commutator subgroup - the group you need to quotient by to make SL(2,Z)
be abelian. This subgroup has finite cohomological dimension and its
Euler characteristic is -1. I'm not sure why this is true, but Brown says
so! This means the Euler characteristic of SL(2,Z) works out to be -1/12.
If you've read my stuff about Euler characteristics in "week147", you'll
see why this gets me so excited - I can add this stuff to my list of
weird ways of calculating the Euler characteristic. Plus, it's related
to the magical role of the number "24" in string theory, and also the
Riemann zeta function!
Indeed, the Riemann zeta function gives a way to make rigorous Euler's
zany observation that
1 + 2 + 3 + .... = -1/12,
as I explained here:
9) John Baez, Euler's Proof that 1+2+3+ ... = -1/12, Bernoulli
Numbers and the Riemann Zeta Function, Winter 2004 Quantum Gravity
Seminar, homework for weeks 5,6,7, available at
http://math.ucr.edu/home/baez/qg-winter2004/
This suggests that there should be a version of the Eilenberg-Mac Lane
space for SL(2,Z) which has 1 cell of dimension 0, 2 cells of dimension
2, 3 cells of dimension 4, and so on. Does anyone know if this is true?
More generally, G. Harder computed the (generalized) Euler characteristic
for a large class of arithmetic groups:
10) G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined
groups, Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455.
For example, he looked at the symplectic group defined over the
integers, Sp(n,Z), and showed that its Euler characteristic is
zeta(-1) zeta(-3) ... zeta(1-2n)
In the case n = 1 we get back SL(2,Z) and zeta(-1) = -1/12.
In fact, for every Chevalley group over the integers, Harder computed its
Euler characteristic in terms of Bernoulli numbers. A Chevalley group is
sort of like a simple Lie group, but defined algebraically. For example,
the Euler characteristic of the integral form of the exceptional group E7
is some wacky number like
- 691 x 43867 / 2^{21} x 3^9 x 5^2 x 7^3 x 11 x 13 x 19
Serre went even further, computing Euler characteristics of Chevalley
groups defined over algebraic number fields. He also noticed that when
you write the Euler characteristic of a group as a fraction, the primes
in the denominator are precisely the primes p for which the group has
p-torsion. He was thus able to conclude, for example, that E7 defined
over the integers has p-torsion for p = 2, 3, 5, 7, 11, 13 and 19.
For more, see:
11) Jean-Pierre Serre, Cohomologie des groups discretes, Ann. Math.
Studies 70 (1971), 77-169.
This only takes us up to 1971. I shudder to think what bizarre
results along these lines are known by now! Probably they'd seem not
bizarre but beautiful if I understood this stuff better: I don't really
have a clue how the Riemann zeta function gets into this game, so
everything after that seems like black magic to me - bewitching but
bewildering.
But, it's clear that the study of groups and symmetry has not lost its
ability to turn up surprises.
Quotes of the Week:
"Regarding the fundamental investigations of mathematics, there is
no final ending ... no first beginning." - Felix Klein
"In point of fact, it has traditionally been the "continuous" aspect of
things which has been the central focus of Geometry, while those properties
associated with "discreteness", notably computational and combinatorial
properties, have been passed over in silence or treated as an afterthought.
It was therefore all the more astonishing to me when I made the discovery,
about a dozen years ago, of the combinatorial theory of the icosahedron,
even though this theory is barely scratched (and probably not even understood)
in the classic treatise of Felix Klein on the icosahedron. I see in this
another significant indicator of this indifference (of over 2000 years)
of geometers vis-a-vis those discrete structures which present themselves
naturally in Geometry: observe that the concept of the group (notably of
symmetries) appeared only in the last century (introduced by Evariste
Galois), in a context that was considered to have nothing to do with
Geometry. Even in our own time it is true that there are lots of
algebraists who still haven't understood that Galois theory is primarily,
in essence, a geometrical vision, which was able to renew our understanding
of so-called "arithmetical" phenomena." - Alexander Grothendieck
-----------------------------------------------------------------------
Addenda: I had often wondered how Klein's name got attached to the
pathetic little "4-group" mentioned above, which is just Z/2 x Z/2.
John McKay proffered an explanation:
There is a group called the Klein group.
It is denoted V_4 = The Vierer-Gruppe (The fours group).
Klein worked with the simple group of order 168 and found
the "Klein quadric" which has it for symmetry group.
The suggestion is that friends decided to call the non-cyclic
abelian group of order 4 the "Klein group" = the "little group"
as a joke.
I have a question you may like to posit to your readers:
Is V_4 the abstract group or a permutation group?
There are other points ... I presume you know that your
-1/12 is zeta(-1). There is a paper by Lepowsky on the occurrence
of such zeta(-n) involving vertex algebras.
I dearly wish I understood cohomology!
I am busy tethering moonshine!
Best,
John
This group of order 168 has made an appearance here before, in "week194":
it's PSL(3,Z/2) - the group of symmetries of the projective plane over
Z/2, or "Fano plane", whose points can also be thought of as imaginary
unit octonions. It's also PSL(2,Z/7). I've long been mystified by its
relation to Klein's quartic, mainly because I've never spent time trying
to understand it! - it's just one of those things that's been gnawing at
the edges of my consciousness, especially when I saw this book come out:
12) Silvio Levy, The Eightfold Way: the Beauty of Klein's Quartic Curve,
MSRI Research Publications 35, Cambridge U. Press, Cambridge 1999.
Review available at http://www.msri.org/publications/books/Book35/
It has a translation of Klein's original paper on this subject.
Someday I'll break down and study this.
Anyway....
James Dolan mentioned some other folklore saying that the "Kleinian
groups" were *also* named after Klein as a joke:
by the way, i enjoyed the latest twf a lot (although i don't know why
we seem to never get a chance to talk about all this stuff ourselves
that much), but i noticed that you (apparently non-ironically)
mentioned kleinian groups as a famous thing named after klein, without
telling the story that i always hear about how poincare gave kleinian
groups the name "kleinian groups" after klein complained to poincare
about poincare's use of the terminology "fuchsian groups" for
something that fuchs apparently didn't event.
i guess that the versions of the story that i'd heard seemed to
suggest that klein was complaining because he thought that fuchs
hadn't significantly contributed to the study of fuchsian groups, and
that poincare may have been naively trying to placate klein and/or
not-so-naively twitting him by then giving the name "kleinian groups"
to something that klein hadn't significantly contributed to the study
of.
however i did just look for the story on the web, and the tellings
that i found there i guess don't really suggest that klein didn't
"significantly contribute to the study of" kleinian groups (or at
least not by my standards). it's still not clear though what sort of
reaction poincare may have been trying to provoke in klein, and
whether he succeeded in provoking it. it's claimed that poincare did
come up with the name "kleinian function" later in the same day after
klein complained about the name "fuchsian function", and also that
klein was subsequently just as vociferous in complaining to poincare
about the name "kleinian function" as he was in complaining about the
name "fuchsian function". but apparently klein's complaints were
based on _very_ exacting concerns about absolute priority, so that the
names "fuchsian function" and "kleinian function" can be seen as
inappropriate only by the standards of someone with similarly
ridiculous concerns about absolute priority, rather than by a
reasonable person such as myself.
i'd also heard that klein's nervous breakdown was provoked by the
stress of trying to keep up with a genius like poincare, but maybe it
was actually provoked by poincare's apparently casual attitude towards
priority disputes and/or concept-naming.
i'd thought of asking you about this issue of whether klein really did
have much to do with kleinian groups right after i read the advance
copy of twf that you sent me, but i guess that i didn't notice that it
was an advance copy. i guess that it doesn't matter though, since
apparently there _is_ a case to be made that klein had lots to do with
developing the theory of kleinian groups; just not by his own
apparently ridiculous standards.
Noam Elkies suggested that the commutator subgroup of SL(2,Z)
has Euler characteristic -1 because it's a a free group on 2
generators, so its classifying space is a figure 8, with Euler
characteristic 1 - 2 = -1 since it has one vertex and two edges.
This sounds right. In particular, I already mentioned how Brown
claims the commutator subgroup of SL(2,Z) is torsion-free. Further,
Kevin Buzzard shows below that any torsion-free subgroup of SL(2,Z)
is a free group. So, we just need to check that the commutator
subgroup of SL(2,Z) can be generated by two elements but not by just one.
Laurent Bartholdi just made this job easier; he sent me an email saying
these are free generators for the commutator subgroup of SL(2,Z):
( 2 -1)
(-1 1)
and
( 1 -1)
(-1 2)
In fact, Kevin Buzzard's email was packed with wisdom:
I know one elementary argument which you don't appear to, so I thought
I'd fill you in. The argument below is waffly but rather easy really.
> But, SL(2,Z) has a torsion-free subgroup of index 12, namely its
> commutator subgroup - the group you need to quotient by to make SL(2,Z)
> be abelian. This subgroup has finite cohomological dimension and its
> Euler characteristic is -1. I'm not sure why this is true, but Brown says
> so! This means the Euler characteristic of SL(2,Z) works out to be -1/12.
One doesn't have to use such a "strange" subgroup as the commutator
subgroup of SL_2(Z). People who do modular forms, like me, far
prefer "congruence subgroups", as these are the ones that show
up when you study automorphic forms for SL_2. So here's an easy
way to compute the Euler characteristic of SL_2(Z): take your
favourite congruence subgroup which has no torsion, work out
its Euler characteristic (this is easy, I'll show you how to do it
in a second) and then deduce what the Euler characteristic of SL_2(Z) is.
Here are some examples of congruence subgroups: for any integer N>=1,
consider the subgroup Gamma_1(N) of SL_2(Z), defined as the matrices
(a b;c d) in SL_2(Z) such that c=0 mod N and a=d=1 mod N. It's just
the pre-image in SL_2(Z) of the upper triangular unipotent matrices
in SL_2(Z/NZ) so it's a subgroup of SL_2(Z). Here's a neat fact
that makes life easy:
Lemma: if N>=5 then Gamma_1(N) has no torsion.
Proof: say g in SL_2(Z) has finite order d>=1. Then its min poly
divides X^d-1 so over the complexes it has distinct linear factors
so it's diagonalisable with roots of unity z and w on the diagonal.
Now |z|=|w|=1 so |trace(g)|<=2. But it's an integer, so it's -2,-1,0,1,2.
And for N>=5 the only one of these congruent to 2 mod N is 2. So z=w=1
and so g is the identity.
Deeper, but also completely standard (and not logically necessary
for what follows)---any torsion-free subgroup
of SL_2(Z) is free! This is because SL_2(Z) acts very naturally
on a certain tree in the upper half plane. This is a neat piece
of mathematics. SL_2(Z) acts on the upper half plane {z=x+iy:y>0}
via the rule (a b;c d) sends z to (az+b)/(cz+d). Now draw dots
at the points i=sqrt(-1) and rho=exp(2*pi*i/6), the primitive 6th
root of unity in the upper half plane, and draw the obvious arc
between them (the one that lies on the circle |z|=1), this is
our first edge, and now look at the image of what you have
under the SL_2(Z) action. It's a rather pretty tree, with two kinds
of vertices---those in the i orbit have valency 2 and stabiliser
of order 4, and those in the rho orbit have valency 3 and stabiliser
of order 6. Now a group is free iff it acts freely on a tree,
and anything torsion-free in SL_2(Z) must be acting freely
on this tree because the stabiliser of each vertex and edge under
the SL_2(Z) action is finite.
So Gamma_1(5) is, by this general theorem, free. In fact I don't
really need this general nonsense, one can give a hands-on proof
of this fact, which I'll do now. We've seen that SL_2(Z), and hence
Gamma_1(5), acts on the upper half plane. There is no torsion
in Gamma_1(5) so the action is very nice, one checks easily that
the action is free in fact by a similar sort of argument to the lemma
above, it's the sort of thing you can find in the first few pages
of any book on modular forms. So we can quotient out the upper half
plane by Gamma_1(5) and get a quotient Riemann surface. The point
is that this computation is very manageable and can be done "in practice".
There is a standard argument which shows how to quotient out the
upper half plane by SL_2(Z)---the answer is a Riemann surface
isomorphic to the complex plane (although you have to take care
at the points where the action isn't free---this is exactly the vertices
of the tree above), and the isomorphism can even
be given "explicitly" via the j-function coming from the theory
of elliptic curves---there is a standard fundamental domain even,
the one with corners rho, rho^2 and +i*infinity. I'm sure you'll have
come across this sort of thing many times before. Now SL_2(Z)
surjects onto SL_2(Z/5Z) so the index of Gamma_1(5) in SL_2(Z)
is just the index of (1 *;0 1) in SL_2(Z/5Z) and by counting
orders this comes out to be 24. Now it's not hard to find explicitly
24 translates of the standard fundamental domain and then glue
them together to work out the quotient of the upper half plane
by Gamma_1(5)---it turns out that it is isomorphic to the
Riemann Sphere minus 4 points.
In fact there is no need to do this sort of computation---the
modular forms people have automated it long ago. The quotient
of the upper half plane by Gamma_1(N) is a Riemann surface
called Y_1(N) and I can just ask my computer to compute the
genus of its natural compactification (this exists and is called X_1(N))
and also to compute how many cusps were added to compactify it.
So in practice you just have to find a friendly modular forms person and
then say "hey, what's the genus of X_1(5) and how many cusps
does it have?" and then you have a complete description of Gamma_1(5)
because it's pi_1 of the answer.
OK, the upper half plane modulo Gamma_1(5) is the sphere minus 4 points,
so Gamma_1(5) is pi_1 of this, i.e. it's free on three generators.
That makes the Euler Characteristic of Gamma_1(5) equal to 1-3=-2.
And we already checked that the index was 24, so the Euler Characteristic
of SL_2(Z) works out to be -1/12.
Grothendieck wouldn't have chosen Gamma_1(5); he would have chosen
something called Gamma(2), the subgroup of SL_2(Z) consisting
of the matrices which are the identity mod 2. There is another
classical modular function lambda inducing an isomorphism
of Y(2), the quotient of the upper half plane by Gamma(2),
with the sphere minus three points---this is what gives
the one-line proof of the fact that any analytic function C-->C
that misses two points must be constant, because it then lifts
to a function from C to the upper half plane which is the same
as the unit disc, so we're done by Liouville. There is a subtlety
here though: (-1 0;0 -1) is in Gamma(2). So you have to work
with PSL_2(Z)=SL_2(Z)/(+-1) instead. Let PGamma(2) denote the image
of Gamma(2) in PSL_2(Z). Note that -1 is kind of a pain in the
theory of modular forms sometimes because it acts trivially on
everything but isn't the identity. Grothendieck was very interested
in the sphere minus three points but it's much older than this
that PGamma(2) is its fundamental group, so PGamma(2) has
Euler characteristic 2-3=-1 and index 6 in PSL_2(Z), so PSL_2(Z)
has Euler characteristic -1/6, so SL_2(Z) has Euler characteristic
-1/12 because that's how they work :-)
> This only takes us up to 1971. I shudder to think what bizarre
> results along these lines are known by now! Probably they'd seem not
> bizarre but beautiful if I understood this stuff better: I don't really
> have a clue how the Riemann zeta function gets into this game, so
> everything after that seems like black magic to me - bewitching but
> bewildering.
Nowadays almost any analytic function that is involved in number
theory, when evaluated at certain "natural" points, gives
an answer which has a conjectural interpretation in terms
of relations between cohomology theories---this is the
subject of many conjectures (Deligne, Beilinson, Bloch-Kato,...).
It is still absolutely black magic! Actually I'm being unfair,
the relation between special values of zeta and Euler characteristics
is somehow less profound than this stuff. I wish I knew more about it!
It can actually be used to compute certain values of L-functions
(things more general than the zeta function but along the same lines)...
Kevin
I replied:
Hi -
Thanks VERY much for this email. I was actually wondering why
Brown used the commutator subgroup of SL(2,Z) as a kind of "warmup"
for computing the Euler characteristic of SL(2,Z) instead of one of
the congruence subgroups. It seems this subgroup is not any of the
beloved congruence subgroups....
In fact, I've finally managed to turn up the thing I was looking
for. How does this relate to the stuff you're saying? It involves
Gamma(3) rather than the Gamma_1(N) groups:
In "week 97", I wrote:
Where does the extra 24 come from? I don't know, but Stephan Stolz
said it has something to do with the fact that while PSL(2,Z) doesn't
act freely on the upper half-plane - hence these elliptic curves with
extra symmetries - the subgroup "Gamma(3)" does. This subgroup consists
of integer matrices
(a b)
(c d)
with determinant 1 such that each entry is congruent to the corresponding
entry of
(1 0)
(0 1)
modulo 3.
So, if we form
H/Gamma(3)
we get a nice space without any "points of greater symmetry".
To get the moduli space of elliptic curves from this, we just
need to mod out by the group
SL(2,Z)/Gamma(3) = SL(2,Z/3)
But this group has 24 elements!
In fact, I think this is just another way of explaining the
period-24 pattern in the theory of modular forms, but I like
it.
Kevin wrote:
>It's a rather pretty tree,
Yes, there's a picture of it in Brown's book, drawn on top of
an old picture by Klein of a triangulation of the hyperbolic
plane.
What Brown seems to be doing there is showing that this tree
is a deformation retract of that triangulation (with its simplicial
topology, where the points on the boundary of the hyperbolic plane
form a discrete set), and thus proving that the cohomological dimension
of SL(2,Z) is just 1.
Anyway, this is all great stuff. Do you mind if I attach a copy of
your email to the copy of "week213" on my website? I think people
will find it helpful, especially because of its friendly
straight-to-the-point style, which books rarely seem to manage....
Best,
jb
Kevin replied:
John Baez wrote:
> I was actually wondering why
> Brown used the commutator subgroup of SL(2,Z) as a kind of "warmup"
> for computing the Euler characteristic of SL(2,Z) instead of one of
> the congruence subgroups. It seems this subgroup is not any of the
> beloved congruence subgroups....
You're right, I don't think it is. For N>=1 define Gamma(N) to
be the kernel of the obvious map SL(2,Z)-->SL(2,Z/NZ); a congruence
subgroup is any subgroup of SL(2,Z) that contains one of these
Gamma(N)'s. Clearly such things have finite index in SL(2,Z). But
unfortunately there exist subgroups of finite index in SL(2,Z) that
are not congruence subgroups. This is a "low-dimensional"
phenomenon---the moment you have a bit more freedom, e.g. you're
working with SL(3,Z) or indeed SL(n,Z) for any n>=3, or even
SL(2,Z[1/p]) for some prime p, then any subgroup of finite index
is a congruence subgroup---these groups satisfy the "congruence
subgroup property". But I've never understood the commutator
of SL(2,Z) precisely for the reason that it's not a congruence
subgroup (this is essentially because the commutator subgroup
of SL(2,Z/NZ) never has index 12 in SL(2,Z/NZ)! The index is always
smaller than 12 because SL(2,Z/pZ) is essentially a simple group.)
> In fact, I've finally managed to turn up the thing I was looking
> for. How does this relate to the stuff you're saying? It involves
> Gamma(3) rather than the Gamma_1(N) groups.
Anything will do. If you know about Gamma(3) then great. The same
key observation is true---Gamma(3) contains no elements of finite
order, because any finite order element (a b;c,d) of Gamma(3) which isn't
the identity must have trace in {-2,-1,0,1} congruent to 2 mod 3,
so the trace must be -1, so d=-1-a, so the det is a(-1-a) mod 9,
which is never 1 mod 9. Now the index of Gamma(3) in SL_2(Z) is 24,
and the modular curve X(3) has genus 0 (everyone knows this because Wiles
needed it to prove Fermat's Last Theorem!) and four cusps (zero, 1, 1/2
and infinity) and hence the Euler Characteristic of Gamma(3) is 2-4=-2, so
we recover the result that the Euler Characteristic of SL_2(Z) is -1/12 again.
John Baez wrote:
> Where does the extra 24 come from? I don't know, but Stephan Stolz
> said it has something to do with the fact that while PSL(2,Z) doesn't
> act freely on the upper half-plane - hence these elliptic curves with
> extra symmetries - the subgroup "Gamma(3)" does.
One can see that any subgroup of SL(2,Z) which has finite index and
is free, must have index a multiple of 12 (and hence at least 12). Because
if it has index d and is free on g generators, when we know (1-g)/d=-1/12,
so 12 divides the denominator of (1-g)/d in lowest terms. Geometrically
what is going on is that perhaps the "correct" quotient of the upper
half plane by SL_2(Z) is not just the complex numbers, it's something that
looks a bit like the complex numbers except there is a little bit of
extra magic going on at i and rho, corresponding to the fact that one
shouldn't really have attempted to quotient out there, one should
just remember that really the quotient is kind of "crumpled up"
near there. So for example the fundamental group of the quotient
shouldn't be the trivial group---if you take a small loop around i then
this should not be regarded as contractible---you have to go around i
twice before you can hope to contract the loop. Similarly you have to
go around rho three times. Even worse---if you do this carefully enough
then even going around i twice or rho three times isn't enough to
contract the loop---because the resulting loop somehow corresponds
to the element -1 in SL_2(Z), which acts trivially but which isn't
the identity! So you have to do everything again before you
get to the element 1. Mumford thought hard about how to make all this
sort of thing rigorous, and managed in the late 60s to prove that the
Picard group of the quotient of the upper half plane by SL_2(Z) was in
fact Z/12Z.
John Baez wrote:
> Anyway, this is all great stuff. Do you mind if I attach a copy of
> your email to the copy of "week213" on my website?
Go ahead!
Kevin
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