# June 18, 2000 {#week150}

First I'd like to say some stuff about Lagrange points. Then I'll
continue talking about complex oriented cohomology theories.

Euler and Lagrange won the Paris Academy Prize in 1722 for their work on
the orbit of the moon. The essay that Lagrange submitted for this prize,
"Essai sur le probleme des trois corps", contained some interesting
results on the 3-body problem. Among other things, he studied the case
where a lighter body was revolving about a heavier one in a circular
orbit, and found all the places where a third much lighter body could
sit "motionless" with respect to the other two. There are 5 such
places, and they're now called the "Lagrange points" L1 -- L5.

For example: imagine the moon going around the earth in a circular
orbit. Then there are 5 Lagrange points where we can put a satellite. 3
of these are unstable equilibria. They lie on the line through the earth
and moon. L1 is between the earth and moon, L2 is in the same direction
as the moon but a bit further out, and L3 is opposite the moon. The
other 2 Lagrange points are stable equilibria. L4 lies in the moon's
orbit 60 degrees ahead of the moon, while L5 lies 60 degrees behind the
moon.

Here's a nice picture of the Lagrange points:

1) "Lagrange points", `http://map.gsfc.nasa.gov/m_mm/ob_techorbit1.html`

In general, the points L4 and L5 will be stable equilibria as long as
the heavy body (e.g. the earth in the above example) is more than 25
times as massive as the intermediate-sized one (e.g. the moon). And in
case you're wondering, this magic number is just an approximation to
the exact figure, which is
$$\frac{25}{2}\left(1+\sqrt{1-\frac{4}{625}}\right) = 24.95993579437711227887\ldots$$
Here's a proof that the Lagrange points work as advertised, including a
derivation of the above number:

2) Neil J. Cornish, "The Lagrange points", available at `http://www.astro.princeton.edu/~njc/lagrange.ps.gz`

Now, Lagrange did his calculation as a mathematical exercise and didn't
believe it was relevant to the actual solar system. But he was wrong
about that! The stable Lagrange points L4 and L5 are quite interesting,
because stuff tends to accumulate there.

For example, people have found over six hundred asteroids called
"Trojans" at the stable Lagrange points of Jupiter's orbit around the
sun. The first to be discovered was 588 Achilles, back in 1906 --- the
number here meaning that it was the 588th asteroid found. In general,
the Trojans at L4 are named after Greek soldiers in the Trojan war,
while those at L5 are named after actual Trojans --- soldiers from the
city of Troy! You can see the Trojans quite clearly in this picture of
the asteroid belt:

3) Asteroid belt, `http://www-groups.dcs.st-and.ac.uk/~history//Diagrams/Asteroids.gif`

The asteroid 5261 Eureka is a "Martian Trojan", occupying the L5 point
of Mars' orbit around the sun. A second Martian Trojan was discovered
at L5 in 1998, but it doesn't have a name yet. For more information,
try:

4) Minor Planet Center, Trojan minor planets, `http://cfa-www.harvard.edu/cfa/ps/lists/Trojans.html`

There may also be a few small asteroids at the Lagrange points of Venus
and Earth's orbits around the sun. Does anyone know more about this? My
brief search revealed only some information about the curious asteroid
3753 Cruithne, which is a companion asteroid of the Earth. But 3753
Cruithne is not at a Lagrange point! Instead, it moves in a very
complicated spiralling horse-shoe shaped orbit relative to the earth.
For a beautiful explanation with pictures by the discoverers of this
phenomenon, see:

5) Paul Wiegert, Kimmo Innanen and Seppo Mikkola, Near-earth asteroid 3753 Cruithne --- Earth's curious companion, `http://www.astro.queensu.ca/~wiegert/`

For over 150 years, astronomers have been searching for other satellites
of the Earth besides the big one I see out my window right now. There
have been a lot of false alarms, with people even giving names to the
satellites they thought they discovered, like "Kleinchen" and
"Lilith". For the fascinating story of these "second moons" and
other hypothetical planets, see:

6) Paul Schlyter, "Hypothetical planets", `http://seds.lpl.arizona.edu/nineplanets/nineplanets/hypo.html`

Unfortunately, none of these second moons were real. But in the 1960s,
people discovered dust clouds at the stable Lagrange points of the
Moon's orbit around the Earth! They are about 4 times as big as the
Moon, but they are not very substantial.

What else lurks at Lagrange points?

Well, Saturn has a moon called Dione, and 60 degrees ahead of Dione,
right at the Lagrange point L4, there is a tiny moon called Helene.
Here's a picture of Helene:

7) Astronomy picture of the day: "Dione's Lagrange moon Helene", `http://antwrp.gsfc.nasa.gov/apod/ap951010.html`

Isn't she cute? There's also a small moon called Telesto at the L4
point of Saturn's moon Tethys, and one called Calypso at the L5 point
of Tethys:

8) Bill Arnett, Introduction to the nine planets: Tethys, Telesto and
Calypso, `http://seds.lpl.arizona.edu/nineplanets/nineplanets/tethys.html`

Does anyone know other natural occupants of Lagrange points?

For a long time crackpots and science fiction writers have talked about
a "Counter-Earth", complete with its own civilization, located at the
L3 point of the Earth's orbit around the Sun --- exactly where we can
never see it! I think satellite explorations have ruled out this
possibility by now, but since L3 is an unstable Lagrange point, it was
never very likely to begin with.

On the other hand, fans of space exploration have long dreamt of setting
up a colony at the stable Lagrange points of the Moon's orbit around
the earth --- right in those dust clouds, I guess. But now people are
putting artificial satellites at the Lagrange points of the Earth's
orbit around the Sun! For example, L1 is the home of "SOHO": the Solar
and Heliospheric Observatory. Sitting between the Earth and Sun gives
SOHO a nice clear view of sunspots, solar flares, and the solar wind.
It's not stable, but it can exert a bit of thrust now and then to stay
put. For more information and some pretty pictures, try:

9) SOHO website, `http://sohowww.nascom.nasa.gov/`

In April 2001, NASA plans to put a satellite called "MAP" at the
Lagrange point L2. MAP is the Microwave Anisotropy Probe, which will
study anisotropies in the cosmic microwave background. These are
starting to be a really interesting window into the early history of the
universe. For more, see:

10) MAP website, `http://map.gsfc.nasa.gov/`

Finally, for quantum analogues of the Lagrange points, see:

11) T. Uzer, Ernestine A. Lee, David Farrelly, and Andrea F. Brunello,
Synthesis of a classical atom: wavepacket analogues of the Trojan
asteroids, Contemp. Phys. 41 (2000), 1-14. Abstract available at
`http://www.catchword.co.uk/titles/tandf/00107514/v41n1/contp1-1.htm`

Okay, enough about Lagrange points! Now I want to talk a bit more about
complex oriented cohomology theories. Last time I left off at the
definition. So let me start with a little review, and then plunge ahead.

A generalized cohomology theory assigns to each space $X$ a bunch of
groups $h^n(X)$, one for each integer $n$. We impose some axioms that make
them work very much like ordinary cohomology. However, when $X$ is a
point, we no longer require that $h^n(X)$ is trivial for nonzero $n$. It
may not seem like much, but it turns out to make a big difference! There
are all sorts of very interesting examples.

Like what?

Well, there's K-theory, which lets us study a space by looking at
vector bundles over that space and its iterated suspensions. We get
various flavors of K-theory from various kinds of vector bundle: real
K-theory, complex K-theory, quaternionic K-theory, and so on. There's
even a sort of K-theory invented by Atiyah that's based on Clifford
algebras, called "KR theory". For more about all these, try:

12) Dale Husemoller, _Fibre Bundles_, Springer-Verlag, New York, 1975.

13) H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, _Spin Geometry_, Princeton University Press, Princeton, 1989.

Then there's cobordism theory, which lets us study a space by looking
at manifolds mapped into that space and its iterated suspensions. To be
honest, this is actually how *bordism* theory works --- this being a
generalized *homology* theory. But every generalized homology theory
goes hand-in-hand with a generalized cohomology theory, and if you
understand one you understand the other, at least in principle....
Anyway, there are various flavors of cobordism theory corresponding to
various kinds of extra structure you can put on a manifold:
piecewise-linear cobordism theory, smooth cobordism theory, oriented
cobordism theory, spin cobordism theory, complex cobordism theory,
symplectic cobordism theory, stable homotopy theory, and so on. For more
about these, try:

14) Robert E. Stong, _Notes on Cobordism Theory_, Princeton University Press, Princeton, 1968.

Finally, there are generalized cohomology theories inspired more by
algebra than by geometry, which only hardcore homotopy theorists seem to
understand, like Morava K-theory and Brown-Peterson theory.

To round off this little tour, I guess I should add that there are lots
of maps going between different generalized cohomology theories! As I
explained in ["Week 149"](#week149), each generalized cohomology
$h^n$ corresponds to a "spectrum": a list of spaces, each being the
loop space of the next. Spectra form a category, and given a map between
spectra we get a map between their generalized cohomology theories. So
we shouldn't study these one of at a time: it's better to play around
with lots at once! For some important examples of the stuff you can do
this way, try:

15) P. E. Conner and E. E. Floyd, _The Relation of Cobordism to K-theories_, Lecture Notes in Mathematics **28**, Springer-Verlag, New York, 1966.

16) Douglas C. Ravenel, _Complex Cobordism and Stable Homotopy Groups of Spheres_, Academic Press, 1986.

and also the books on generalized cohomology listed in ["Week 149"](#week149).

Anyway, to bring order to this zoo, it's nice to pick out some of the
special features of *ordinary* cohomology theory and study the
generalized cohomology theories that share these features.

For example, in ordinary cohomology, the cohomology groups of a space
fit together to form a graded ring. If a generalized cohomology theory
is like this, we call it "multiplicative". Whenever this is the case,
we get a graded ring $R$ called the "coefficient ring" of our theory,
which is simply the cohomology ring of the one-point space. For ordinary
cohomology theory the coefficient ring is just $\mathbb{Z}$, but for other theories
it can be very interesting and complicated. By easy abstract nonsense,
the cohomology ring of any space is an algebra over the coefficient
ring.

Another nice feature of ordinary cohomology is the first Chern class.
Whenever you have a complex line bundle over a space $X$, you get an
invariant called its "first Chern class" which lives in $H^2(X)$, and
this invariant is sufficiently powerful to completely classify such line
bundles. Last week I described the *universal* line bundle over
infinite-dimensional complex projective space:
$$L\to \mathbb{CP}^\infty$$
and showed how the first Chern class of *any* line bundle comes from the
first Chern class of this one, which I called $c$.

If a generalized cohomology theory is multiplicative and there's an
element $c$ of $h^2(\mathbb{CP}^\infty)$ that acts like the first Chern class of the
universal line bundle, we call the theory "complex oriented". Of
course, to make this precise we need to isolate the key features of the
first Chern class and abstract them. I did this in
["Week 149"](#week149), so I won't do it again here. Instead,
I'll just say a bit about what we can *do* with a complex oriented
cohomology theory.

For starters, we can use the element $c$ to get an invariant of complex
line bundles --- a kind of generalized version of the first Chern class.
To do this, just remember from ["Week 149"](#week149) that $\mathbb{CP}^\infty$ is
the classifying space for complex line bundles. In other words, *any*
line bundle over *any* space $X$ is isomorphic to a pullback of the
universal line bundle by some map
$$f\colon X\to\mathbb{CP}^\infty.$$
Thus, given a line bundle we can find such a map $f$ and use it to pull
back the element $c$ to get an element of $h^2(X)$. This is exactly like
the usual first Chern class of our line bundle, except now we're using
a generalized cohomology theory instead of ordinary cohomology.

Can we get any *other* invariants of complex line bundles from *other*
elements of the cohomology of $\mathbb{CP}^\infty$? Not really: in any complex oriented
cohomology theory, the cohomology ring of $\mathbb{CP}^\infty$ is just the algebra of
formal power series in the element $c$:
$$h^*(\mathbb{CP}^\infty) = R[[c]]$$
where $R$ is the coefficient ring.

However, we can get other invariants of complex *vector* bundles, which
are analogous to the higher Chern classes. In fact, we can mimic a huge
amount of the usual theory of characteristic classes in the context of a
complex oriented cohomology theory. I'd love to talk about this, but it
would be a digression from my main goal, which is to make elliptic
cohomology at least halfway comprehensible to the amateur.

So: last time I mentioned that $\mathbb{CP}^\infty$ is an abelian topological group,
with a multiplication map
$$m\colon \mathbb{CP}^\infty \times \mathbb{CP}^\infty\to \mathbb{CP}^\infty$$
and an inverse map
$$i\colon \mathbb{CP}^\infty\to \mathbb{CP}^\infty.$$
And I explained how these represent the operations of *tensoring* two
line bundles and taking the *dual* of a line bundle, respectively. Now
let's see what we can do with these maps when we have a complex
oriented cohomology theory. First of all, since cohomology is
contravariant we get homomorphisms
$$m^*\colon h^*(\mathbb{CP}^\infty)\to h^*(\mathbb{CP}^\infty \times \mathbb{CP}^\infty)$$
and
$$i^*\colon h^*(\mathbb{CP}^\infty)\to h^*(\mathbb{CP}^\infty)$$
But as I already said, we have
$$h^*(\mathbb{CP}^\infty) = R[[c]]$$
and similarly we have
$$h^*(\mathbb{CP}^\infty \times \mathbb{CP}^\infty) = R[[c]] \otimes R[[c]]$$
where the product "$\otimes$" on the right side is a slightly fattened-up
version of the usual tensor product of algebras over $R$. So we really
have homomorphisms
$$m^*\colon R[[c]]\to R[[c]] \otimes R[[c]]$$
$$i^*\colon R[[c]]\to R[[c]]$$
which satisfy all the usual axioms for the product and inverse in an
abelian group --- but turned around backwards.

Folks who like Hopf algebras will immediately note that $R[[c]]$ is
like a Hopf algebra. A nice way to form a Hopf algebra is to take the
algebra of functions on a group and use the product and inverse in the
group to give this algebra extra operations called the "coproduct" and
"antipode". We're doing the same thing here, except that we're using
formal power series in one variable $c$ instead of functions of one
variable. So folks call $R[[c]]$ a "formal group law".

In short: complex oriented cohomology theories give formal group laws!

Lest this seem overly abstract and unmotivated, remember that it's just
a way of talking about what happens to the generalized "first Chern
class" when we tensor line bundles. In a vague but useful way, we can
visualize guys in $R[[c]]$ as formal power series on the line, where
the line has been equipped with some abelian group structure, at least
right near the origin. This group structure is what yields the coproduct
$$m^*\colon R[[c]]\to R[[c]] \otimes R[[c]]$$
and antipode
$$i^*\colon R[[c]]\to R[[c]]$$
But the real point is that this group structure tells us how to compute
the generalized "first Chern class" of a tensor product of line
bundles starting from both of their generalized first Chern classes.

Some examples may help. In ordinary cohomology theory, when we tensor
two line bundles, we just *add* their first Chern classes. So in this
case, we've got the line made into a group using addition, and
$R[[c]]$ becomes a formal group law called the "additive formal group
law".

Another famous example is complex K-theory. In this theory, when we
tensor two line bundles, we basically just *multiply* their generalized
first Chern classes. That's because cohomology classes in K-theory are
just equivalence classes of vector bundles, and multiplying them just
*means* tensoring them. So in this case, $R[[c]]$ becomes the
"multiplicative" formal group law. Of course, some fiddling around is
required, because we don't usually think of the multiplicative unit $1$
as the origin of the line... but we can if we want.

What are some other examples? Well, complex cobordism theory is one.
This corresponds to the "universal" formal group law: a formal group
law that has a unique homomorphism to any other one! In this case $R$ is
quite big: it's called the "Lazard ring". And this universal aspect
of complex cobordism theory makes it the king of all complex oriented
generalized cohomology theories. I really should spend about 10 pages
explaining it to you in detail, but I won't....

... because I want to finally say a word about elliptic cohomology!

I've talked a lot about elliptic curves in ["Week 13"](#week13),
["Week 125"](#week125) and ["Week 126"](#week126), so I get to
assume you know about them: an elliptic curve is a $1$-dimensional compact
complex manifold which is also an abelian group. As such, any elliptic
curve gives a formal group law. And thus we can wonder if this formal
group law comes from a complex oriented cohomology theory... and it
does! And this is elliptic cohomology!

Now this is just the beginning of a long story: there's much more to
say, and I don't have the energy to say it here... but I'll just
tantalize you with some of the high points. Since elliptic curves can be
thought of as the worldsheets of strings, there are a bunch of
interesting relationships between string theory and elliptic cohomology.
In addition to the references I've gave you in
["Week 149"](#week149), see for example:

17) Hirotaka Tamanoi, _Elliptic Genera and Vertex Operator Super-Algebras_, Springer Lecture Notes in Mathematics **1704**, Springer, Berlin, 1999.

We also get lots of nice relationships to the theory of modular forms.
But the one thing we apparently don't have *yet* is a very neat
*geometrical* description of elliptic cohomology! There's got to be
one, since so much nice geometry is involved... but what is it?

(For an answer to this question, see ["Week 197"](#week197).)

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**Addendum:** For the proof that the complex cobordism theory
corresponds to the universal formal group law, read Quillen's original
paper on the subject:

18) Daniel Quillen, "On the formal group laws of unoriented and complex cobordism theory", _Bull. Amer. Math. Soc._ **75** (1969), 1293--1298. Also available as `http://projecteuclid.org/euclid.bams/1183530915`

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> *In any field, find the strangest thing and then explore it.*
> 
> --- John Wheeler