Also available at http://math.ucr.edu/home/baez/week286.html
December 8, 2009
This Week's Finds in Mathematical Physics (Week 286)
John Baez
This Week I'd like to start telling you about "rational homotopy
theory". But first: can you guess what this is a picture of?
http://math.ucr.edu/home/baez/abalos_undae.jpg
I'll explain it at the end.
So, what's "rational homotopy theory"? One might naively define it
as the study of spaces whose homotopy groups are vector spaces over
the rational numbers.
But if you think about it, that's pretty weird!
For example, the first homotopy group of a space X, usually called the
"fundamental group" of X and denoted pi_1(X), consists of equivalence
classes of loops in X that start and end at our favorite point. Two
loops count as equivalent if you can continuously deform one until it
looks like the other. If you can do this, we say these loops are
"homotopic".
The fundamental group of the circle is Z, the group of integers.
The reason is that two loops drawn on the circle are homotopic
if and only if they wind around the same number of times - and
that number must be an integer! You can walk around the block once
and get back home. You can walk around the block twice and get back
home. You can even walk around the block -5 times and get back
home: the negative number just means you walk around the other way.
But you can't walk *halfway* around the block and be back home!
But suppose you had a space whose fundamental group was Q, the
rational numbers. Then you *could* walk halfway around the block and
get back home. That sounds pretty weird - nay, downright impossible!
But part of why it sounds so weird is that it's not right. We really
need some other "block" such that walking around *that* block *twice*
is homotopic to walking around the original block *once*. This sounds
more complicated... but also more possible.
Later in this post I'll describe a space called "the rational circle",
whose fundamental group is indeed Q. Then you can see how it actually
works.
Anyway: spaces whose homotopy groups are rational vector spaces are
weird. Why should we care about them?
It's better not to think of rational homotopy theory as the study
of weird spaces whose homotopy groups are rational vector spaces.
It's better to think of it as the study of *ordinary* spaces -
but viewed in a way that doesn't let us see their homotopy
groups, only their homotopy groups tensored with Q. This process
turns their homotopy groups into rational vector spaces!
This is a common theme in algebraic topology. We can think of
various kinds of homotopy theory either as the completely precise
study of rather strange spaces, or as the study of ordinary spaces
as seen through a blurry lens. A blurry lens can be a good thing,
because it simplifies a complicated picture.
However, even *this* way of thinking about rational homotopy theory
misses the point. The real point is that rational vector spaces come
from the land of linear algebra, so rational homotopy blends topology
and linear algebra. So does rational *homology* theory, but rational
homotopy theory is deeper. When we get into it, we'll take lots of
important concepts from linear algebra - like commutative algebras,
and Lie algebras, and Hopf algebras - and study very interesting
"homotopy versions" of these concepts.
By doing this, we'll vastly generalize linear algebra. We'll wind up
with a whole new perspective... and we'll see applications to physics
ranging from classical field theory, to quantization, to supergravity!
And you should not be surprised that we're doing here is really
*categorifying* linear algebra.
But more on that later. Today, I want to start with the naive
viewpoint that rational homotopy theory is about spaces whose homotopy
groups are rational vector spaces.
In algebraic topology, the really hard part is *torsion*. A group
element is "torsion" if you can add it to itself a bunch of times
and get zero. So, for example, every element of a finite group
is torsion, but the group of integers is "torsion-free".
Look at some homotopy groups of spheres and you'll see what I mean:
pi_3(S^2) = Z
pi_5(S^3) = Z/2
pi_7(S^4) = Z x Z/12
pi_9(S^5) = Z/2
pi_11(S^6) = Z
pi_13(S^7) = Z/2
pi_15(S^8) = Z x Z/120
pi_17(S^9) = Z/8
pi_19(S^10) = Z x Z/8
These are the homotopy groups pi_{2n-1}(S^n). If you were asked to
make a guess about the torsion-free part of these groups, you could
easily formulate a conjecture: it's Z when n is even, and trivial
when n is odd. And this is true.
But if you were asked to make a guess about the torsion part of these
groups, you'd find it a lot harder. And indeed, nobody knows the full
story here.
This suggests trying to do a version of algebraic topology where we
systematically get rid of torsion. We'll lose a lot of important
information, but things will get easy and fun - and still far from
trivial!
This is "rational homotopy theory".
How can we get rid of torsion?
Well, the nth homotopy group of a compact manifold, like a sphere, is
always finitely generated - and abelian when n > 1. A finitely
generated abelian group always looks like Z^n x T where T is finite.
All the torsion is in T, so to get rid of torsion we can just throw
out T.
But that doesn't work in general. In general, the nth homotopy group
of a space can be *any* group when n = 1 - and any abelian group when
n > 1.
For an arbitrary abelian group, the torsion elements always form a
subgroup, called the "torsion subgroup". It's not true in general
that an abelian group is the product of its torsion subgroup and
some other group! But, we can still kill off the torsion by modding
out by the torsion subgroup.
For a nonabelian group, the torsion elements don't necessarily form
a subgroup! For example, take the free group generated by x and y,
and mod out by the relations x^2 = y^2 = 1. Then x and y are torsion
elements, but xy is not.
I don't know any good way to kill off the torsion for an arbitrary
nonabelian group. A lot of work on rational homotopy theory sidesteps
this issue by working only with "1-connected" spaces. These are
spaces that are path-connected and also simply connected. That means the
fundamental group is trivial - and the higher homotopy groups are
always abelian, so we don't have to worry about nonabelian groups.
Now, I've made it sound like the right way to "kill off torsion" in an
abelian group is to mod out by its torsion subgroup. This makes me
wonder if there's a systematic way to take a space X and turn it into
a space X' such that pi_n(X') is pi_n(X) mod its torsion subgroup.
Does anyone know?
But anyway, this is *not* how we kill off torsion in rational homotopy
theory!
Instead, here's what we do. Abelian groups are the same as Z-modules
where Z is the ring of integers. Since Z is commutative, we can take
tensor products of Z-modules. In other words, we can take tensor
products of abelian groups. And to kill off the torsion in an abelian
group, we just tensor it with the rational numbers!
I hope you see what this accomplishes. Tensoring an abelian group G
with the rational numbers gives a new abelian group Q tensor G that
includes elements like
q tensor g
where g is in G and q is a rational number. Any element g of G gives
an element of Q tensor G, namely
1 tensor g
But we also get elements like
(1/2) tensor g
which acts like "half of g". More generally, given any element of Q
tensor G, we're allowed to multiply it by any fraction.
Now, suppose g is a torsion element of G. Then ng = 0 for some n, so
1 tensor ng = 0,
If we multiply both sides by 1/n, we get
1 tensor g = 0
So, torsion elements of G get sent to zero in Q tensor G. We've killed
the torsion.
But the great thing about this trick is that Q tensor G is even better
than a torsion-free abelian group. It's a vector space over the
rational numbers! So, we're not just killing off torsion. We're
going from the world of abelian groups to the world of *linear
algebra*, which is notoriously well-behaved.
Next let me sketch how we can take a 1-connected space X and
"rationalize" it, obtaining a new space X_Q with
pi_n(X_Q) = pi_n(X) tensor Q
for all n.
Since we're doing homotopy theory, we can assume X is a "CW complex".
A space of this sort is built from balls. To build a CW complex, we
start with some 0-balls - that is, points. Then we take some 1-balls
- that is, intervals - and glue their boundaries to the 0-balls. We
get a space that's just a graph. Then we take some 2-balls - that is,
disks - and glue their boundaries to the space we've got so far. Then
we take some 3-balls and glue their boundaries to what we've got so
far. And so on, ad infinitum. Any space is "weakly homotopy
equivalent" to a space of this sort, and that's good enough for us.
So, to rationalize X we should rationalize this whole procedure!
This procedure relies on balls - and also spheres, since the
boundary of a ball is a sphere. So, we should define a "rational
n-ball" and a "rational n-sphere", and then make sure that given
a CW complex, we can build a new space where each ball or sphere we
used has been replaced by a "rational" one!
I'll describe the rational n-sphere, since that's the fun part.
Even though we don't need it here, let's start with the case n = 1:
the "rational circle". As mentioned earlier, this is a space whose
fundamental group is Q. Here's one way to build it.
First, take an ordinary circle, and take a cylinder, and glue the
circle to the bottom of the cylinder. But: make sure the circle goes
around the bottom of the cylinder *twice*! See what this
accomplishes? It means that walking around your original circle
*once* is homotopic to walking around the top of the cylinder 2 times.
This solves our problem of how walking once around the block can be
the same as walking twice around some other block.
Then take another cylinder, and glue the top of your first cylinder
to that. But: make sure the top of your first cylinder winds around
the bottom of this new one *3 times*.
Then take yet another cylinder. Glue the top of your second
cylinder to the bottom of that - but make sure it wraps around
the bottom *4 times*.
And then do this forever...
...and then take a little rest, since it's very tiring to do an
infinite amount of work. Sit back and admire your handiwork. The
space you've built has Q as its fundamental group, because for any
loop g and any integer n, we've created a new loop h such that g = nh.
Mathematicians call this general type of space a "telescope".
An ordinary hand telescope - the kind that pirates use - is built
from cylinders of metal that fit into each other. A mathematician's
telescope is similar, but it's built from infinitely many cylinders,
and you can't collapse it, because they're attached to each other
in a complicated way. This makes it really easy to spot a
mathematician in a roomful of pirates.
We can easily generalize this telescope idea to construct the
"rational n-sphere". The point is that for each integer k, there's a
way to wrap the n-sphere around itself k times. So, we can use these
to build an infinite telescope, just as we did for the rational
circle. This telescope is a space whose homotopy groups are those of
the n-sphere, but tensored with the rational numbers.
A similar trick produces a rational n-ball, but this is less exciting,
since all the homotopy groups of the n-ball were trivial already -
it's contractible, after all. The rational n-ball is still
contractible, but it's been modified so that its "boundary" is a
rational n-sphere.
Having rationalized our spheres and balls, we also need to check
that the maps we used to build our CW complex extend in a canonical
way from the spheres to the rational spheres. But let's skip
this: in This Week's Finds we only do the fun part!
As you can see, the rationalized version of a nice little CW complex
is usually a huge nightmarish space. This is a familiar tradeoff in
algebra topology: spaces with comprehensible homotopy groups almost
always look big and scary when we try to build them by gluing balls
together. But it's a tradeoff algebraic topologists have learned to
accept. There's more to life than whether a space *looks* nice.
In particular, this rationalization process has a very nice abstract
characterization. Suppose X is any 1-connected pointed space. Then
we can define "a rationalization" of X to be any 1-connected pointed
space X' equipped with a map
X -> X'
satisfying two properties. First, X' is a "rational space": a
1-connected pointed space whose homotopy groups are rational
vector spaces. Second, the induced map
Q tensor pi_n(X) -> Q tensor pi_n(X')
is isomorphism for all n.
It turns out that the rationalization of a space is unique up to weak
homotopy equivalence. And we can construct it for CW complexes as I
just explained.
Okay. So far I've been treating rational homotopy theory as the study
of weird "rational" spaces. And I've showed you how to turn any space
into one of these. But as I already admitted, this misses the point.
To come closer to the point, we need to recall an amazing old
theorem due to J. H. C. Whitehead, which says a map
f: X -> Y
between connected CW complexes is a homotopy equivalence if and only
if the induced maps
pi_n(f): pi_n(X) -> pi_n(Y)
are isomorphisms for all n. This is why for more general connected
spaces we define any map that induces isomorphisms on homotopy groups
to be a "weak homotopy equivalence". Even better, every space is
weakly homotopy equivalent to a CW complex! So, if we formally throw
in inverses to all weak homotopy equivalences, we get a category
called where every space is *isomorphic* to a CW complex. This is
called the "homotopy category".
These ideas are very powerful, so it's good to generalize them to
rational homotopy theory. So now suppose X and Y are 1-connected
pointed spaces. And let's say a map
f: X -> Y
is a "rational homotopy equivalence" if the induced maps
on rational homotopy groups
Q tensor pi_n(f): Q tensor pi_n(X) -> Q tensor pi_n(Y)
are isomorphisms for all n. There's a nice category where we formally
throw in inverses to all rational homotopy equivalences. This is
called the "rational homotopy category".
In the rational homotopy category, we're studying ordinary spaces
through a blurry lens, because two spaces that have a rational
homotopy equivalence between them look the same.
But the rational homotopy category is equivalent to a subcategory of
the usual homotopy category: namely, the subcategory consisting of
rational spaces and all morphisms between those! So, we can also
say we're studying strange spaces, but with complete precision - or
at least, the usual level of precision in homotopy theory.
To learn more, I urge you to grab this and take a look:
1) Kathryn Hess, Rational homotopy theory: a brief introduction,
in Interactions Between Homotopy Theory and Algebra, ed.
Luchezar L. Avramov, Contemp. Math 436, AMS, Providence, Rhode
Island, 2007. Also available as arXiv:math.AT/0604626.
For even more detail, I recommend:
2) Yves Felix, Stephen Halperin and Jean-Claude Thomas, Rational
Homotopy Theory, Springer, Berlin, 2005.
I'll give more references later. In the weeks to come, we'll see two
very different descriptions of the rational homotopy category, which
were both discovered by Daniel Quillen back in 1969:
3) Daniel Quillen, Rational homotopy theory, Ann. Math. 90 (1969),
205-295.
It's these other descriptions that make the subject really interesting.
One is based on a homotopy version of Lie algebras. The other is
based on a homotopy version of commutative algebras!
In the first sentence of his paper, Quillen defines the rational
homotopy category. But he does it a bit differently than I just did.
This is worth understanding. He says it's "the category obtained from
the category of 1-connected pointed spaces by localizing with respect
to the family of maps which are isomorphisms modulo the class in the
sense of Serre of torsion abelian groups".
Let me say this with less jargon. Start with the category of
1-connected pointed spaces. Throw in formal inverses of all maps
f: X -> Y
for which the induced maps
pi_n(f): pi_n(X) -> pi_n(Y)
have kernels and cokernels where all elements are torsion. This gives
the rational homotopy category!
I'll conclude with a few remarks that would have been a bit too
distracting earlier.
First: I discussed rational homotopy theory only for 1-connected
spaces. This is great as far as the connection to algebra goes. But
in terms of topology it's a bit sad. Sometimes people go a step
further and work with "nilpotent" spaces. These are spaces where the
fundamental group is nilpotent and acts nilpotently on the higher
homotopy groups.
Second: the rational circle is an interesting space. As we've seen,
it's a space with the rational numbers as its fundamental group. All
its other homotopy groups are trivial, since that's already true for
the circle.
Any space with G as its nth homotopy group and every other homotopy
group being trivial is called "the Eilenberg-Mac Lane space K(G,n)".
We're allowed use the word "the", since this space is unique up to
weak homotopy equivalence. So, the rational 1-sphere is K(Q,1).
I've talked about lots of different Eilenberg-Mac Lane spaces in
This Week's Finds, and they're all collected here:
3) John Baez, Generalized cohomology theories, Eilenberg-Mac Lane
spaces, classifying spaces and K(Z,n), and other examples of
classifying spaces. Available at
http://math.ucr.edu/home/baez/calgary/BG.html
Now you can add K(Q,1) to your collection!
Third: in case you're wondering about Quillen's jargon: by
"localizing" he means the process of taking a category and throwing in
formal inverses to a bunch of morphisms. This is an important way of
simplifying categories. It lets us count slightly different objects
as the same.
A "Serre class" of abelian groups is a bunch of abelian groups such
that whenever A and C are in this class, and the sequence
0 -> A -> B -> C -> 0
is exact, then B is in this class too. The idea is that we think of
abelian groups in the Serre class as "small", or "negligible". For
example: the class of finite abelian groups, or the class of torsion
abelian groups. We can localize the category of abelian groups by
throwing in an inverse for any morphism whose kernel and cokernel are
in the Serre class.
If you like abelian categories, you can generalize this "Serre
class" idea from the category of abelian groups to other abelian
categories.
There's also much more to say about localization! Try this for
starters:
4) nLab, Localization, http://ncatlab.org/nlab/show/localization
Besides doing rational homotopy theory, we can use localization to
take homotopy theory and "localize at the prime p". This is a way to
focus special attention on the "p-torsion" in our homotopy groups:
that is, the elements that give zero when you multiply them by a power
of p.
Finally, what about the picture at the beginning of this Week's Finds?
It shows sand dunes in a region called Abalos Undae near the north
pole of Mars:
5) HiRISE (High Resolution Imaging Science Experiments),
Dunes in Abalos Undae, http://hirise.lpl.arizona.edu/PSP_010219_2785
The photo covers a strip about 1.2 kilometers across. As the HiRISE
satellite sweeps over Mars it takes incredibly detailed photos like
this. Here's the description on the HiRISE website:
The Abalos Undae dune field stretches westward, away from a
portion (Abalos Colles) of the ice-rich north polar layered
deposits that is separated from the main Planum Boreum dome by
two large chasms. These dunes are special because their sands
may have been derived from erosion of the Rupes Tenuis unit (the
lowest stratigraphic unit in Planum Boreum, beneath the icier
layers) during formation of the chasms. Some researches have
argued that these chasms were formed partially by melting of the
polar ice.
The enhanced color data illuminate differences in composition.
The dunes appear blueish because of their basaltic composition,
while the reddish-white areas are probably covered in dust.
Upon close inspection, tiny ripples and grooves are visible on
the surface of the dunes; these features are formed by wind
action, as are the dunes themselves.
It is possible that the dunes are no longer migrating (the
process of dune formation forces dunes to move in the direction
of the main winds) and that the tiny ripples are the only active
parts of the dunes today.
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Quote of the Week:
"...the pursuit of science is more than the pursuit of understanding.
It is driven by the creative urge, the urge to construct a vision, a
map, a picture of the world that gives the world a little more beauty
and coherence than it had before. Somewhere in the child that urge
is born." - John Archibald Wheeler
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Addenda: For more discussion visit the n-Category Cafe, here:
http://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_47.html
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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
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