John Baez
Oct12-06, 04:10 AM
Also available as http://math.ucr.edu/home/baez/week220.html
August 31, 2005
This Week's Finds in Mathematical Physics - Week 220
John Baez
Work on quantum gravity has seemed stagnant and stuck for the
last couple of years, which is why I've been turning more
towards pure math.
Over in string theory they're contemplating a vast "landscape" of
possible universes, each with their own laws of physics - one or
more of which might be ours. Each one is supposed to correspond
to a different "vacuum" or "background" for the marvelous unifying
M-theory that we don't completely understand yet. They can't
choose the right vacuum except by the good old method of fitting
the experimental data. But these days, this time-honored method
gets a lot less airplay than the "anthropic principle":
1) Leonard Susskind, The anthropic landscape of string theory,
available as hep-th/0302219.
Perhaps this is because it's more grandiose to imagine choosing
one theory out of a multitude by discovering that it's among the
few that supports intelligent life, than by noticing that it
correctly predicts experimental results. Or, perhaps it's because
nobody really knows how to get string theory to predict experimental
results! Even after you chose a vacuum, you'd need to see how
supersymmetry gets broken, and this remain quite obscure.
There's still tons of beautiful math coming out of string theory,
mind you: right now I'm just talking about physics.
What about loop quantum gravity? This line of research has always
been less ambitious than string theory. Instead of finding the
correct theory of everything, its goal has merely been to find
*any* theory that combines gravity and quantum mechanics in a
background-free way. But, it has major problems of its own:
nobody knows how it can successfully mimic general relativity
at large length scales, as it must to be realistic! Old-fashioned
perturbative quantum gravity failed on this score because it
wasn't renormalizable. Loop quantum gravity may get around this
somehow... but it's about time to see exactly how.
Loop quantum gravity follows two main approaches: the so-called
"Hamiltonian" or "spin network" approach, which focuses on the
geometry of space at a given time, and the so-called "Lagrangian"
or "spin foam" approach, which focuses on the geometry of
spacetime.
In the last couple of years, the most interesting new work in the
Hamiltonian approach has focussed on problems with extra symmetry,
like black holes and the big bang. Here's a nontechnical
introduction:
2) Abhay Ashtekar, Gravity and the quantum, available as
gr-qc/0410054.
and here's some new work that treats the information loss
puzzle:
3) Abhay Ashtekar and Martin Bojowald, Black hole evaporation:
a paradigm, Class. Quant. Grav. 22 (2005) 3349-3362. Also
available as gr-qc/0504029.
However, by focusing on solutions with extra symmetry, one puts
off facing the hardest aspects of renormalization, or whatever
its equivalent might be in loop quantum gravity.
The other approach - the spin foam approach - got stalled when
the most popular model seemed to give spacetimes made mostly of
squashed-flat "degenerate 4-simplexes". Various papers have
found an effect like this: see "week198" for more details. So,
there's definitely a real phenomenon going on here. However,
its physical significance remains a bit obscure. The devil is
in the details.
In particular, even though the *amplitude* for a single large
4-simplex in the Barrett-Crane model is dominated by degenerate
geometries, certain *second derivatives* of the amplitude might not -
and this may be what really matters. Carlo Rovelli has recently
come out with a paper on this:
4) Carlo Rovelli, Graviton propagator from background-independent
quantum gravity, available as gr-qc/0508124.
If the idea holds up, I'll be pretty excited. If not, I'll be
bummed. But luckily, I've already gone through the withdrawal
pains of switching my focus away from quantum gravity. When you
do theoretical physics, sometimes you feel the high of discovering
hidden truths about the physical universe. Sometimes you feel the
agony of suspecting that those "hidden truths" were probably just
a bunch of baloney... or, realizing that you may never know.
Ultimately nature has the last word.
Math is (at least for me) a less nerve-racking pursuit, since
the truths we find can be confirmed simply by discussing them:
we don't need to wait for experiment. Math is just as grand as
physics, or more so. But it's more wispy and ethereal, since it's
about pure pattern in general - not the particular magic patterns
that became the world we see. So, the stakes are lower, but the
odds are higher.
Speaking of math, I really want to talk about the Streetfest - the
conference in honor of Ross Street's 60th birthday. It was a real
blast: over sixty talks in two weeks in two cities, Sydney and
Canberra. However, I accidentally left my notes from those talks
at home before zipping off to Calgary for a summer school on
homotopy theory:
5) Topics in Homotopy Theory, graduate summer school at the
Pacific Institute of Mathematics run by Kristine Bauer and Laura
Scull. Recommended reading material available at
http://www.pims.math.ca/science/2005/05homotopy/reading.html
So, I'll say a bit about what I learned at this school.
Dan Dugger spoke about motivic homotopy theory, which was
*great*, because I've been trying to understand stuff
from number theory and algebraic geometry like the Weil
conjectures, etale cohomology, motives, and Voevodsky's proof
of the Milnor conjecture... and thanks to his wonderfully
pedagogical lectures, it's all starting to make some sense!
I hope to talk about this someday, but not now.
Alejandro Adem spoke about orbifolds and group cohomology.
Purely personally, the most exciting thing here was seeing
that orbifolds can also be seen as certain kinds of topological
groupoids, or stacks, or topoi... so that various versions of
"categorified topology" are actually different faces of the
same thing!
I may talk about this someday, too, but not now.
I spoke about higher gauge theory and its relation to Eilenberg-
Mac Lane spaces. I may talk about that too someday, but not now.
Dev Sinha spoke about operads, and besides explaining the basics,
he said a couple of things that really blew me away. So, I want
to talk about this now.
For one, the homology of the little k-cubes operad is a graded
version of the Poisson operad! For two, the little 2-cubes
operad acts on the space of thickened long knots!
But for this to thrill you like it thrills me, I'd better say a
word about operads - and especially little k-cubes operads.
Operads, and especially the little k-cubes operads, were
invented by Peter May in the early 1970s to formalize the
algebraic structures lurking in "infinite loop spaces". In
"week149" I explained what infinite loop spaces are, and how
they give generalized cohomology theories, but let's not get
bogged down in this motivation now, since operads are actually
quite simple.
In its simplest form, an operad is a gizmo that has for each
n = 0,1,2,... a set O(n) whose elements are thought of as n-ary
operations - operations with n inputs. It's good to draw such
operations as black boxes with n input wires and one output:
\ | /
\ | /
\ | /
-----
| f |
-----
|
|
For starters these operations are purely abstract things that
don't actually operate on anything. Only when we consider a
"representation" or "action" of an operad do they get incarnated
as actual n-ary operations on some set. The point of operads is
to study their actions.
But, for completeness, let me sketch the definition of an operad.
An operad tells us how to compose its operations, like this:
\ / \ | / |
\ / \ | / |
----- ----- -----
| b | | c | | d |
----- ----- -----
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
-----
| a |
-----
|
|
Here we are composing a with b,c, and d to get an operation with 6
inputs called a o (b,c,d).
An operad needs to have a unary operation serving as the identity
for composition. It also needs to satisfy an "associative law"
that makes a composite of composites like this well-defined:
\ / | \ | / \ /
\ / | \ | / \ /
--- --- --- ---
| | | | | | | |
--- --- --- ---
\ | / /
\ | / /
\ | / /
----- ----- -----
| | | | | |
----- ----- -----
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
-----
| |
-----
|
|
(This picture has a 0-ary operation in it, just to emphasize
that this is allowed.)
That's the complete definition of a "planar operad". In a
full-fledged operad we can do more: we can permute the inputs
of any operation and get a new operation:
\ / /
/ /
/ \ /
/ /
/ / \
\ | /
-----
| |
-----
|
|
This gives actions of the permutation groups on the sets O(n).
We also demand that these actions be compatible with composition,
in a way that's supposed to be obvious from the pictures. For
example:
\ | / | \ / \\\ / / /
\ | / | \ / \\/ / /
--- --- --- /\\ / /
| a | | b | | c | / \\/ /
--- --- --- / / /
\ / / / / /\\
\ / / / | | \\\
\ / / / | | \\\
/ / --- --- ---
/ \ / = | b | | c | | a |
/ / --- --- ---
/ / \ \ | /
\ | / \ | /
----- -----
| d | | d |
----- -----
| |
| |
and similarly for permuting the inputs of the black boxes on
top.
Voila!
Now, operads make sense in various contexts. So far we've been
talking about operads that have a *set* O(n) of n-ary operations
for each n. These have actions on *sets*, where each guy in O(n)
gets incarnated as a *function* that eats n elements of some set
and spits out an element of that set.
But historically, Peter May started by inventing operads that have
a *topological space* of n-ary operations for each n. These like
to act on *topological spaces*, with the operations getting
incarnated as *continuous maps*.
Most importantly, he invented an operad called the "little k-cubes
operad". Here O(n) is the space of ways of putting n disjoint
little k-dimensional cubes in a big one. We don't demand that
the little cubes are actually cubes: they can be rectangular
boxes. We do demand that their walls are nicely lined up with
the walls of the big cube:
---------------------
| |
| ----- |
| ----- | | |
|| | | | |
|| | | | | typical
| ----- | | | 3-ary operation in the
| ----- | little 2-cubes operad
| ---------------- |
| | | |
| ---------------- |
| |
---------------------
This is an operation in O(3), where O is the little 2-cubes
operad. Or, at least it would be if I labelled each of the 3
little 2-cubes - we need that extra information.
We compose operations by sticking pictures like this into
each of the little k-cubes in another picture like this!
I should draw you an example, but I'm too lazy. So, figure
it out yourself and check the associative law.
The reason this example is so important is that we get an action
of the little k-cubes operad whenever we have a "k-fold loop
space".
Starting from a space S equipped with a chosen point *, the
k-fold loop space Omega^k(S) is the space of all maps from
a k-sphere into S that send the north pole to the point *. But
this is also the space of all maps from a k-cube into S sending
the boundary of the k-cube to the point *.
So, given n such such maps, we can glom them together using an
n-ary operation in the little k-cubes operad:
---------------------
|*********************|
|***********-----*****|
|*-----****| |****|
|| |***| |****|
|| |***| |****|
| ----- ***| |****|
|***********-----*****|
|***----------------**|
|**| |*|
|***----------------**|
|*********************|
---------------------
where we map all the shaded stuff to the point *. We get
another map from the k-cube to S sending the boundary to *.
So:
ANY k-FOLD LOOP SPACE HAS AN ACTION OF
THE LITTLE k-CUBES OPERAD!
But the really cool part is the converse:
ANY CONNECTED POINTED SPACE WITH AN ACTION OF
THE LITTLE k-CUBES OPERAD IS
HOMOTOPY EQUIVALENT TO A k-FOLD LOOP SPACE!
This is too technical to make a good bumper sticker, so if you
want people in your neighborhood to get interested in operads,
I suggest combining both the above slogans into one:
A k-FOLD LOOP SPACE IS THE SAME AS
AN ACTION OF THE LITTLE k-CUBES OPERAD!
Like any good slogan, this leaves out some important fine print,
but it gets the basic idea across. Modulo some details, being a
k-fold loop space amounts to having a bunch of operations: one
for each way of stuffing little k-cubes in a big one!
By the way:
Speaking of bumper stickers, I'm in Montreal now, and there's
a funky hangout on the Boulevard Saint-Laurent called Cafe Pi
where people play chess - and they sell T-shirts, key rings,
baseball caps and coffee mugs decorated with the Greek letter pi!
The T-shirts are great if you're going for a kind of math-nerd/
punk look; I got one to wow the students in my undergraduate
courses. I don't usually provide links to commercial websites,
but I made an exception for Acme Klein Bottles, and I'll make an
exception for Cafe Pi:
5) Cafe Pi, http://www.cafepi.ca/
Unfortunately they don't sell bumper stickers.
But where were we? Ah yes - the little k-cubes operad.
The little k-cubes operad sits in the little (k+1)-cubes operad
in an obvious way. Indeed, it's a "sub-operad". So, we can
take the limit as k goes to infinity and form the "little
infinity-cubes operad". Any infinite loop space gets an action
of this... and that's why Peter May invented operads!
You can read more about these ideas in May's book:
6) J. Peter May, The Geometry of Iterated Loop Spaces,
Lecture Notes in Mathematics 271, Springer, Berlin, 1972.
or for a more gentle treatment, try this expository article:
7) J. Peter May, Infinite loop space theory, Bull. Amer. Math.
Soc. 83 (1977), 456-494.
But Dev Sinha told us about some subsequent work by Fred
Cohen, who computed the homology and cohomology of the little
k-cubes operad.
For this, we need to think about operads in the world of linear
algebra. Here we consider operads that have a *vector space* of
n-ary operations for each n, which get incarnated as *multilinear
maps* when they act on some *vector space*. These are sometimes
called "linear operads".
An example is the operad for Lie algebras. This one is called
"Lie". Lie(n) is the vector space of n-ary operations that one
can do whenever one has a Lie algebra. In this example:
Lie(0) is zero-dimensional, since there are no nullary operations
(constants) built into the definition of Lie algebra.
Lie(1) is one-dimensional, since the only unary operations are
multiples of the identity operation:
a |-> a
Lie(2) is one-dimensional, since the only binary operations are
multiples of the Lie bracket:
(a,b) |-> [a,b]
You might think we need a second guy in Lie(2), namely
(a,b) |-> [b,a]
but the antisymmetry of the Lie bracket says this is linearly
dependent on the first one:
[b,a] = -[a,b]
Lie(3) is two-dimensional, since the only ternary operations
are multiples of these two:
(a,b,c) |-> [[a,b],c]
(a,b,c) |-> [b,[a,c]]
You might think we need a third guy in Lie(3), for example
(a,b,c) |-> [a,[b,c]]
but the Jacobi identity says this is linearly dependent on the
first two:
[a,[b,c]] = [[a,b],c] + [b,[a,c]]
You may enjoy trying to show that the dimension of Lie(n) is
(n-1)!, at least for n > 0. There's an incredibly beautiful
conceptual proof, and probably lots of obnoxious brute-force
proofs.
There's a lot more to say about the Lie operad, but right now
I want to talk about the Poisson operad. A "Poisson algebra"
is a commutative associative algebra that has a bracket operation
{a,b} making it into a Lie algebra, with the property that
{a,bc} = {a,b}c + b{a,c}
So, bracketing with any element is like taking a derivative: it
satisfies the product rule.
For this reason, Poisson algebras arise naturally as algebras of
observables in classical mechanics - the Poisson bracket of any
observable A with an observable H called the "Hamiltonian" tells
you the time derivative of A:
dA/dt = {H,A}
This is the beginning of a nice big story.
But, what's got me excited now is how Poisson algebras show up in
topology!
To understand this, we need to note that there's a linear operad
whose algebras are Poisson algebras. That's not surprising. But,
we can get a very similar operad in a rather shocking way, as
follows.
Take the little k-cubes operad. This has a space O(n) of n-ary
operations for each n. Now take the homology of these spaces
O(n), using coefficients in your favorite field, and get vector
spaces H(O(n)). By functorial abstract nonsense these form a
linear operad. And this is the operad for Poisson algebras!
Alas, we actually have to be a bit more careful. The homology of
O(n) with coefficients in some field is really a *graded* vector
space over that field. So, H(O(n)) really forms an operad in the
category of graded vector spaces. And, it's the operad whose
algebras are graded Poisson algebras with a bracket of degree k-1.
What's that? Well, it's like a Poisson algebra, but if a is an
element of degree |a| and b is an element of degree |b|, then:
ab has degree |a| + |b| (we've got a graded algebra)
{a,b} has degree |a| + |b| + k - 1 (with a bracket of degree k-1)
and the usual axioms for a Poisson algebra hold, but sprinkled
with minus signs according to the usual yoga of graded vector
spaces.
So: whenever we have a k-fold loop space, its homology is a graded
Poisson algebra with a bracket of degree k-1.
To get an idea of this works, let me sketch how the product and
the bracket work. Suppose we have an space X with an action of
the little k-cubes operad:
The product on homology corresponds to sticking two little cubes
side by side. Given two points in X, this gives another point in
X. More generally, given two homology classes a and b in X, we
get a homology class of degree |a| + |b| in X.
The bracket comes from taking one little cube and moving it around
to trace out a sphere surrounding the other little cube. Given
two points in X, this gives a (k-1)-sphere in X. More generally,
given a homology class a in X, and a homology class b in X, we
get a homology class {a,b} of degree |a| + |b| + k - 1.
The equation
{a,bc} = {a,b}c + b{a,c}
then says "moving a around b and c is like moving a around b while
c stands by, plus moving a around c while b stands by".
I guess this result can be found here:
8) Frederick Cohen, Homology of Omega^{n+1}Sigma^{n+1}X and C_{n+1}X,
n > 0, Bull. Amer. Math. Soc. 79 (1973), 1236-1241.
9) Frederick Cohen, Tom Lada and J. Peter May, The homology of
iterated loop spaces, Lecture Notes in Mathematics 533, Springer,
Berlin, 1976.
But, I don't think these old papers talk about graded Poisson
operads! Dev Sinha has a paper where he takes these ideas and
distills them all into the combinatorics of graphs and trees:
10) Dev Sinha, A pairing between graphs and trees, available
as math.QA/0502547.
However, what I really like is how he gets these graphs and
trees starting from the homology and cohomology (respectively)
of the little k-cubes operad! This seems to be lurking in here:
11) Dev Sinha, Manifold theoretic compactifications of
configuration spaces, available as math.GT/0306385
But, I think (and hope) he's writing an expository article that
will explain everything as simply as he did in his lectures!
I have a vague feeling that this relation between the little
k-cubes operad and the Poisson operad is part of a big picture
involving braids and quantization. Another hint in this direction
is Deligne's Conjecture, now proved in many ways, which says that
the operad of singular chains coming from the little 2-disks
operad acts on the Hochschild cochain complex of any associative
algebra. Since Hochschild cohomology classifies the ways you
can deform an associative algebra, this result is related to
quantization and Poisson algebras. But, I don't get the big
picture! This might help:
12) Maxim Kontsevich, Operads and motives in deformation
quantization, Lett. Math. Phys. 48 (1999) 35-72. Also available
as math.QA/9904055.
I'd like to ponder this now! But I'm getting tired, and I still
need to say how the little 2-cubes operad acts on the space of
thickened long knots.
What's a thickened long knot? In k dimensions, it's an embedding
of a little k-cube in a big one:
f: [0,1]^k -> [0,1]^k
subject to the condition that the top and bottom of the little
cube get mapped to the top and bottom of the big one via the
identity map. So, you should imagine a thickened long knot as
a fat square rope going from the ceiling to the floor, all tied
up in knots.
There are two ways to "compose" thickened long knots.
If you're a knot theorist, the obvious way is to stick one on top
of the other - just like the usual composition of tangles. But if
you just think of thickened long knots as functions, you can also
compose them just by composing functions! This amounts to
stuffing one knot inside another... a little hard to visualize,
but fun.
Anyway, it turns out that the whole little 2-cubes operad acts
on the space of thickened long knots, with the two operations
I just mentioned corresponding to this:
---------------------
| |
| |
| |
| |
| | sticking one thickened long
--------------------- knot on top of another
| |
| |
| |
| |
| |
---------------------
and this:
---------------------
| | |
| | |
| | |
| | |
| | | sticking one thickened long
| | | knot inside another
| | |
| | |
| | |
| | |
| | |
---------------------
This isn't supposed to make obvious sense, but the point is, there
are lots of binary operations interpolating between these two -
one for each binary operation in the little 2-cubes operad!
This gives a new proof that the operation of "sticking one
thickened long knot on top of another" is commutative up to
homotopy.
And, using these ideas, Ryan Budney has managed to figure out a
lot of information about the homotopy type of the space of long
knots. Check out these papers:
13) Ryan Budney, Little cubes and long knots, available as
math.GT/0309427.
14) Ryan Budney and Frederick Cohen, On the homology of the space
of long knots, available as math.GT/0504206.
15) Ryan Budney, Topology of spaces of knots in dimension 3,
available as math.GT/0506524.
The paper by Budney and Cohen combines the two ideas I just
described - the action of the little 2-cubes operad on thickened
long knots and its relation to the Poisson operad. Using these,
they show that the rational homology of the space of thickened
long knots in 3 dimensions is a free Poisson algebra! They also
show that the mod-p homology of this space is a free "restricted
Poisson" algebra.
-----------------------------------------------------------------------
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/twf.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
August 31, 2005
This Week's Finds in Mathematical Physics - Week 220
John Baez
Work on quantum gravity has seemed stagnant and stuck for the
last couple of years, which is why I've been turning more
towards pure math.
Over in string theory they're contemplating a vast "landscape" of
possible universes, each with their own laws of physics - one or
more of which might be ours. Each one is supposed to correspond
to a different "vacuum" or "background" for the marvelous unifying
M-theory that we don't completely understand yet. They can't
choose the right vacuum except by the good old method of fitting
the experimental data. But these days, this time-honored method
gets a lot less airplay than the "anthropic principle":
1) Leonard Susskind, The anthropic landscape of string theory,
available as hep-th/0302219.
Perhaps this is because it's more grandiose to imagine choosing
one theory out of a multitude by discovering that it's among the
few that supports intelligent life, than by noticing that it
correctly predicts experimental results. Or, perhaps it's because
nobody really knows how to get string theory to predict experimental
results! Even after you chose a vacuum, you'd need to see how
supersymmetry gets broken, and this remain quite obscure.
There's still tons of beautiful math coming out of string theory,
mind you: right now I'm just talking about physics.
What about loop quantum gravity? This line of research has always
been less ambitious than string theory. Instead of finding the
correct theory of everything, its goal has merely been to find
*any* theory that combines gravity and quantum mechanics in a
background-free way. But, it has major problems of its own:
nobody knows how it can successfully mimic general relativity
at large length scales, as it must to be realistic! Old-fashioned
perturbative quantum gravity failed on this score because it
wasn't renormalizable. Loop quantum gravity may get around this
somehow... but it's about time to see exactly how.
Loop quantum gravity follows two main approaches: the so-called
"Hamiltonian" or "spin network" approach, which focuses on the
geometry of space at a given time, and the so-called "Lagrangian"
or "spin foam" approach, which focuses on the geometry of
spacetime.
In the last couple of years, the most interesting new work in the
Hamiltonian approach has focussed on problems with extra symmetry,
like black holes and the big bang. Here's a nontechnical
introduction:
2) Abhay Ashtekar, Gravity and the quantum, available as
gr-qc/0410054.
and here's some new work that treats the information loss
puzzle:
3) Abhay Ashtekar and Martin Bojowald, Black hole evaporation:
a paradigm, Class. Quant. Grav. 22 (2005) 3349-3362. Also
available as gr-qc/0504029.
However, by focusing on solutions with extra symmetry, one puts
off facing the hardest aspects of renormalization, or whatever
its equivalent might be in loop quantum gravity.
The other approach - the spin foam approach - got stalled when
the most popular model seemed to give spacetimes made mostly of
squashed-flat "degenerate 4-simplexes". Various papers have
found an effect like this: see "week198" for more details. So,
there's definitely a real phenomenon going on here. However,
its physical significance remains a bit obscure. The devil is
in the details.
In particular, even though the *amplitude* for a single large
4-simplex in the Barrett-Crane model is dominated by degenerate
geometries, certain *second derivatives* of the amplitude might not -
and this may be what really matters. Carlo Rovelli has recently
come out with a paper on this:
4) Carlo Rovelli, Graviton propagator from background-independent
quantum gravity, available as gr-qc/0508124.
If the idea holds up, I'll be pretty excited. If not, I'll be
bummed. But luckily, I've already gone through the withdrawal
pains of switching my focus away from quantum gravity. When you
do theoretical physics, sometimes you feel the high of discovering
hidden truths about the physical universe. Sometimes you feel the
agony of suspecting that those "hidden truths" were probably just
a bunch of baloney... or, realizing that you may never know.
Ultimately nature has the last word.
Math is (at least for me) a less nerve-racking pursuit, since
the truths we find can be confirmed simply by discussing them:
we don't need to wait for experiment. Math is just as grand as
physics, or more so. But it's more wispy and ethereal, since it's
about pure pattern in general - not the particular magic patterns
that became the world we see. So, the stakes are lower, but the
odds are higher.
Speaking of math, I really want to talk about the Streetfest - the
conference in honor of Ross Street's 60th birthday. It was a real
blast: over sixty talks in two weeks in two cities, Sydney and
Canberra. However, I accidentally left my notes from those talks
at home before zipping off to Calgary for a summer school on
homotopy theory:
5) Topics in Homotopy Theory, graduate summer school at the
Pacific Institute of Mathematics run by Kristine Bauer and Laura
Scull. Recommended reading material available at
http://www.pims.math.ca/science/2005/05homotopy/reading.html
So, I'll say a bit about what I learned at this school.
Dan Dugger spoke about motivic homotopy theory, which was
*great*, because I've been trying to understand stuff
from number theory and algebraic geometry like the Weil
conjectures, etale cohomology, motives, and Voevodsky's proof
of the Milnor conjecture... and thanks to his wonderfully
pedagogical lectures, it's all starting to make some sense!
I hope to talk about this someday, but not now.
Alejandro Adem spoke about orbifolds and group cohomology.
Purely personally, the most exciting thing here was seeing
that orbifolds can also be seen as certain kinds of topological
groupoids, or stacks, or topoi... so that various versions of
"categorified topology" are actually different faces of the
same thing!
I may talk about this someday, too, but not now.
I spoke about higher gauge theory and its relation to Eilenberg-
Mac Lane spaces. I may talk about that too someday, but not now.
Dev Sinha spoke about operads, and besides explaining the basics,
he said a couple of things that really blew me away. So, I want
to talk about this now.
For one, the homology of the little k-cubes operad is a graded
version of the Poisson operad! For two, the little 2-cubes
operad acts on the space of thickened long knots!
But for this to thrill you like it thrills me, I'd better say a
word about operads - and especially little k-cubes operads.
Operads, and especially the little k-cubes operads, were
invented by Peter May in the early 1970s to formalize the
algebraic structures lurking in "infinite loop spaces". In
"week149" I explained what infinite loop spaces are, and how
they give generalized cohomology theories, but let's not get
bogged down in this motivation now, since operads are actually
quite simple.
In its simplest form, an operad is a gizmo that has for each
n = 0,1,2,... a set O(n) whose elements are thought of as n-ary
operations - operations with n inputs. It's good to draw such
operations as black boxes with n input wires and one output:
\ | /
\ | /
\ | /
-----
| f |
-----
|
|
For starters these operations are purely abstract things that
don't actually operate on anything. Only when we consider a
"representation" or "action" of an operad do they get incarnated
as actual n-ary operations on some set. The point of operads is
to study their actions.
But, for completeness, let me sketch the definition of an operad.
An operad tells us how to compose its operations, like this:
\ / \ | / |
\ / \ | / |
----- ----- -----
| b | | c | | d |
----- ----- -----
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
-----
| a |
-----
|
|
Here we are composing a with b,c, and d to get an operation with 6
inputs called a o (b,c,d).
An operad needs to have a unary operation serving as the identity
for composition. It also needs to satisfy an "associative law"
that makes a composite of composites like this well-defined:
\ / | \ | / \ /
\ / | \ | / \ /
--- --- --- ---
| | | | | | | |
--- --- --- ---
\ | / /
\ | / /
\ | / /
----- ----- -----
| | | | | |
----- ----- -----
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
-----
| |
-----
|
|
(This picture has a 0-ary operation in it, just to emphasize
that this is allowed.)
That's the complete definition of a "planar operad". In a
full-fledged operad we can do more: we can permute the inputs
of any operation and get a new operation:
\ / /
/ /
/ \ /
/ /
/ / \
\ | /
-----
| |
-----
|
|
This gives actions of the permutation groups on the sets O(n).
We also demand that these actions be compatible with composition,
in a way that's supposed to be obvious from the pictures. For
example:
\ | / | \ / \\\ / / /
\ | / | \ / \\/ / /
--- --- --- /\\ / /
| a | | b | | c | / \\/ /
--- --- --- / / /
\ / / / / /\\
\ / / / | | \\\
\ / / / | | \\\
/ / --- --- ---
/ \ / = | b | | c | | a |
/ / --- --- ---
/ / \ \ | /
\ | / \ | /
----- -----
| d | | d |
----- -----
| |
| |
and similarly for permuting the inputs of the black boxes on
top.
Voila!
Now, operads make sense in various contexts. So far we've been
talking about operads that have a *set* O(n) of n-ary operations
for each n. These have actions on *sets*, where each guy in O(n)
gets incarnated as a *function* that eats n elements of some set
and spits out an element of that set.
But historically, Peter May started by inventing operads that have
a *topological space* of n-ary operations for each n. These like
to act on *topological spaces*, with the operations getting
incarnated as *continuous maps*.
Most importantly, he invented an operad called the "little k-cubes
operad". Here O(n) is the space of ways of putting n disjoint
little k-dimensional cubes in a big one. We don't demand that
the little cubes are actually cubes: they can be rectangular
boxes. We do demand that their walls are nicely lined up with
the walls of the big cube:
---------------------
| |
| ----- |
| ----- | | |
|| | | | |
|| | | | | typical
| ----- | | | 3-ary operation in the
| ----- | little 2-cubes operad
| ---------------- |
| | | |
| ---------------- |
| |
---------------------
This is an operation in O(3), where O is the little 2-cubes
operad. Or, at least it would be if I labelled each of the 3
little 2-cubes - we need that extra information.
We compose operations by sticking pictures like this into
each of the little k-cubes in another picture like this!
I should draw you an example, but I'm too lazy. So, figure
it out yourself and check the associative law.
The reason this example is so important is that we get an action
of the little k-cubes operad whenever we have a "k-fold loop
space".
Starting from a space S equipped with a chosen point *, the
k-fold loop space Omega^k(S) is the space of all maps from
a k-sphere into S that send the north pole to the point *. But
this is also the space of all maps from a k-cube into S sending
the boundary of the k-cube to the point *.
So, given n such such maps, we can glom them together using an
n-ary operation in the little k-cubes operad:
---------------------
|*********************|
|***********-----*****|
|*-----****| |****|
|| |***| |****|
|| |***| |****|
| ----- ***| |****|
|***********-----*****|
|***----------------**|
|**| |*|
|***----------------**|
|*********************|
---------------------
where we map all the shaded stuff to the point *. We get
another map from the k-cube to S sending the boundary to *.
So:
ANY k-FOLD LOOP SPACE HAS AN ACTION OF
THE LITTLE k-CUBES OPERAD!
But the really cool part is the converse:
ANY CONNECTED POINTED SPACE WITH AN ACTION OF
THE LITTLE k-CUBES OPERAD IS
HOMOTOPY EQUIVALENT TO A k-FOLD LOOP SPACE!
This is too technical to make a good bumper sticker, so if you
want people in your neighborhood to get interested in operads,
I suggest combining both the above slogans into one:
A k-FOLD LOOP SPACE IS THE SAME AS
AN ACTION OF THE LITTLE k-CUBES OPERAD!
Like any good slogan, this leaves out some important fine print,
but it gets the basic idea across. Modulo some details, being a
k-fold loop space amounts to having a bunch of operations: one
for each way of stuffing little k-cubes in a big one!
By the way:
Speaking of bumper stickers, I'm in Montreal now, and there's
a funky hangout on the Boulevard Saint-Laurent called Cafe Pi
where people play chess - and they sell T-shirts, key rings,
baseball caps and coffee mugs decorated with the Greek letter pi!
The T-shirts are great if you're going for a kind of math-nerd/
punk look; I got one to wow the students in my undergraduate
courses. I don't usually provide links to commercial websites,
but I made an exception for Acme Klein Bottles, and I'll make an
exception for Cafe Pi:
5) Cafe Pi, http://www.cafepi.ca/
Unfortunately they don't sell bumper stickers.
But where were we? Ah yes - the little k-cubes operad.
The little k-cubes operad sits in the little (k+1)-cubes operad
in an obvious way. Indeed, it's a "sub-operad". So, we can
take the limit as k goes to infinity and form the "little
infinity-cubes operad". Any infinite loop space gets an action
of this... and that's why Peter May invented operads!
You can read more about these ideas in May's book:
6) J. Peter May, The Geometry of Iterated Loop Spaces,
Lecture Notes in Mathematics 271, Springer, Berlin, 1972.
or for a more gentle treatment, try this expository article:
7) J. Peter May, Infinite loop space theory, Bull. Amer. Math.
Soc. 83 (1977), 456-494.
But Dev Sinha told us about some subsequent work by Fred
Cohen, who computed the homology and cohomology of the little
k-cubes operad.
For this, we need to think about operads in the world of linear
algebra. Here we consider operads that have a *vector space* of
n-ary operations for each n, which get incarnated as *multilinear
maps* when they act on some *vector space*. These are sometimes
called "linear operads".
An example is the operad for Lie algebras. This one is called
"Lie". Lie(n) is the vector space of n-ary operations that one
can do whenever one has a Lie algebra. In this example:
Lie(0) is zero-dimensional, since there are no nullary operations
(constants) built into the definition of Lie algebra.
Lie(1) is one-dimensional, since the only unary operations are
multiples of the identity operation:
a |-> a
Lie(2) is one-dimensional, since the only binary operations are
multiples of the Lie bracket:
(a,b) |-> [a,b]
You might think we need a second guy in Lie(2), namely
(a,b) |-> [b,a]
but the antisymmetry of the Lie bracket says this is linearly
dependent on the first one:
[b,a] = -[a,b]
Lie(3) is two-dimensional, since the only ternary operations
are multiples of these two:
(a,b,c) |-> [[a,b],c]
(a,b,c) |-> [b,[a,c]]
You might think we need a third guy in Lie(3), for example
(a,b,c) |-> [a,[b,c]]
but the Jacobi identity says this is linearly dependent on the
first two:
[a,[b,c]] = [[a,b],c] + [b,[a,c]]
You may enjoy trying to show that the dimension of Lie(n) is
(n-1)!, at least for n > 0. There's an incredibly beautiful
conceptual proof, and probably lots of obnoxious brute-force
proofs.
There's a lot more to say about the Lie operad, but right now
I want to talk about the Poisson operad. A "Poisson algebra"
is a commutative associative algebra that has a bracket operation
{a,b} making it into a Lie algebra, with the property that
{a,bc} = {a,b}c + b{a,c}
So, bracketing with any element is like taking a derivative: it
satisfies the product rule.
For this reason, Poisson algebras arise naturally as algebras of
observables in classical mechanics - the Poisson bracket of any
observable A with an observable H called the "Hamiltonian" tells
you the time derivative of A:
dA/dt = {H,A}
This is the beginning of a nice big story.
But, what's got me excited now is how Poisson algebras show up in
topology!
To understand this, we need to note that there's a linear operad
whose algebras are Poisson algebras. That's not surprising. But,
we can get a very similar operad in a rather shocking way, as
follows.
Take the little k-cubes operad. This has a space O(n) of n-ary
operations for each n. Now take the homology of these spaces
O(n), using coefficients in your favorite field, and get vector
spaces H(O(n)). By functorial abstract nonsense these form a
linear operad. And this is the operad for Poisson algebras!
Alas, we actually have to be a bit more careful. The homology of
O(n) with coefficients in some field is really a *graded* vector
space over that field. So, H(O(n)) really forms an operad in the
category of graded vector spaces. And, it's the operad whose
algebras are graded Poisson algebras with a bracket of degree k-1.
What's that? Well, it's like a Poisson algebra, but if a is an
element of degree |a| and b is an element of degree |b|, then:
ab has degree |a| + |b| (we've got a graded algebra)
{a,b} has degree |a| + |b| + k - 1 (with a bracket of degree k-1)
and the usual axioms for a Poisson algebra hold, but sprinkled
with minus signs according to the usual yoga of graded vector
spaces.
So: whenever we have a k-fold loop space, its homology is a graded
Poisson algebra with a bracket of degree k-1.
To get an idea of this works, let me sketch how the product and
the bracket work. Suppose we have an space X with an action of
the little k-cubes operad:
The product on homology corresponds to sticking two little cubes
side by side. Given two points in X, this gives another point in
X. More generally, given two homology classes a and b in X, we
get a homology class of degree |a| + |b| in X.
The bracket comes from taking one little cube and moving it around
to trace out a sphere surrounding the other little cube. Given
two points in X, this gives a (k-1)-sphere in X. More generally,
given a homology class a in X, and a homology class b in X, we
get a homology class {a,b} of degree |a| + |b| + k - 1.
The equation
{a,bc} = {a,b}c + b{a,c}
then says "moving a around b and c is like moving a around b while
c stands by, plus moving a around c while b stands by".
I guess this result can be found here:
8) Frederick Cohen, Homology of Omega^{n+1}Sigma^{n+1}X and C_{n+1}X,
n > 0, Bull. Amer. Math. Soc. 79 (1973), 1236-1241.
9) Frederick Cohen, Tom Lada and J. Peter May, The homology of
iterated loop spaces, Lecture Notes in Mathematics 533, Springer,
Berlin, 1976.
But, I don't think these old papers talk about graded Poisson
operads! Dev Sinha has a paper where he takes these ideas and
distills them all into the combinatorics of graphs and trees:
10) Dev Sinha, A pairing between graphs and trees, available
as math.QA/0502547.
However, what I really like is how he gets these graphs and
trees starting from the homology and cohomology (respectively)
of the little k-cubes operad! This seems to be lurking in here:
11) Dev Sinha, Manifold theoretic compactifications of
configuration spaces, available as math.GT/0306385
But, I think (and hope) he's writing an expository article that
will explain everything as simply as he did in his lectures!
I have a vague feeling that this relation between the little
k-cubes operad and the Poisson operad is part of a big picture
involving braids and quantization. Another hint in this direction
is Deligne's Conjecture, now proved in many ways, which says that
the operad of singular chains coming from the little 2-disks
operad acts on the Hochschild cochain complex of any associative
algebra. Since Hochschild cohomology classifies the ways you
can deform an associative algebra, this result is related to
quantization and Poisson algebras. But, I don't get the big
picture! This might help:
12) Maxim Kontsevich, Operads and motives in deformation
quantization, Lett. Math. Phys. 48 (1999) 35-72. Also available
as math.QA/9904055.
I'd like to ponder this now! But I'm getting tired, and I still
need to say how the little 2-cubes operad acts on the space of
thickened long knots.
What's a thickened long knot? In k dimensions, it's an embedding
of a little k-cube in a big one:
f: [0,1]^k -> [0,1]^k
subject to the condition that the top and bottom of the little
cube get mapped to the top and bottom of the big one via the
identity map. So, you should imagine a thickened long knot as
a fat square rope going from the ceiling to the floor, all tied
up in knots.
There are two ways to "compose" thickened long knots.
If you're a knot theorist, the obvious way is to stick one on top
of the other - just like the usual composition of tangles. But if
you just think of thickened long knots as functions, you can also
compose them just by composing functions! This amounts to
stuffing one knot inside another... a little hard to visualize,
but fun.
Anyway, it turns out that the whole little 2-cubes operad acts
on the space of thickened long knots, with the two operations
I just mentioned corresponding to this:
---------------------
| |
| |
| |
| |
| | sticking one thickened long
--------------------- knot on top of another
| |
| |
| |
| |
| |
---------------------
and this:
---------------------
| | |
| | |
| | |
| | |
| | | sticking one thickened long
| | | knot inside another
| | |
| | |
| | |
| | |
| | |
---------------------
This isn't supposed to make obvious sense, but the point is, there
are lots of binary operations interpolating between these two -
one for each binary operation in the little 2-cubes operad!
This gives a new proof that the operation of "sticking one
thickened long knot on top of another" is commutative up to
homotopy.
And, using these ideas, Ryan Budney has managed to figure out a
lot of information about the homotopy type of the space of long
knots. Check out these papers:
13) Ryan Budney, Little cubes and long knots, available as
math.GT/0309427.
14) Ryan Budney and Frederick Cohen, On the homology of the space
of long knots, available as math.GT/0504206.
15) Ryan Budney, Topology of spaces of knots in dimension 3,
available as math.GT/0506524.
The paper by Budney and Cohen combines the two ideas I just
described - the action of the little 2-cubes operad on thickened
long knots and its relation to the Poisson operad. Using these,
they show that the rational homology of the space of thickened
long knots in 3 dimensions is a free Poisson algebra! They also
show that the mod-p homology of this space is a free "restricted
Poisson" algebra.
-----------------------------------------------------------------------
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