The other direction of Ochanine’s theorem

In a previous post, I began discussing a theorem of Ochanine:

Theorem 1 (Ochanine) A genus annihilates the projectivization of every even-dimensional complex bundle if and only if the logarithm of is an elliptic integral

In the previous post, we described Ochanine’s proof that a genus whose logarithm is an elliptic integral (a so-called elliptic genus) annihilated any such projectivization. The proof relied on some computations in the projectivization and then some trickery with elliptic functions. The purpose of this post is to prove the converse: a genus with a suitably large kernel comes from an elliptic integral.

 

The Milnor hypersurfaces

In a previous post, we studied the formal group law of in geometric terms: that is, using the interpretation of as the cobordism ring of stably almost-complex manifolds. We found that the logarithm for this formal group law was given by the power series

In particular, we saw that the complex projective spaces provided a set of independent generators for the rationalization : that is,

This is analogous to the theorem of Hirzebruch which calculates the oriented cobordism ring , and could also have been established directly by arguing that . The structure of the latter ring can be worked out directly, and in fact was.

We might be interested, though, in a set of honest generators for (not generators mod torsion). Such a set is provided by the Milnor hypersurfaces which I would like to discuss in this post.

 

Ochanine’s theorem on elliptic genera

Let be a genus. We might ask when satisfies the following multiplicative property:

Property: For any appropriate fiber bundle of manifolds, we have

When is simply connected, this is true for the signature by an old theorem of Chern, Hirzebruch, and Serre.

A special case of the property (1) is that whenever is an even-dimensional complex vector bundle, then we have

for the projectivization: this is because is a fiber bundle whose fibers are odd-dimensional complex projective spaces, which vanish in the cobordism ring.

Ochanine has given a complete characterization of the genera which satisfy this property.

Theorem 1 (Ochanine) A genus annihilates the projectivizations of even-dimensional complex vector bundles if and only if the associated log series is given by an elliptic integral

for for constants .

Such genera are called elliptic genera. Observe for instance that in the case , then

so that we get the signature as an example of an elliptic genus (the signature has as logarithm, as we saw in the previous post).

I’d like to try to understand the proof of Ochanine’s theorem in the next couple of posts. In this one, I’ll describe the proof that an elliptic genus in fact annihilates projectivizations of even-dimensional bundles .

 

Genera and formal power series

Let be a -algebra. A genus is a homomorphism

where is the oriented cobordism ring. In other words, a genus assigns to every compact, oriented manifold an element . This satisfies the conditions:

1. .
2. .
3. for any manifold-with-boundary .

A fundamental example of a genus is the signature , which assigns to every manifold of dimension the signature of the quadratic form on . (Also, is zero on manifolds whose dimension is not divisible by four.)

 

The Atiyah-Bott fixed point formula

I’d like to take a break from the previous homotopy-theoretic series of posts and do something a bit more geometric here. I’ll describe the classical Atiyah-Bott fixed point formula for an elliptic complex and one of the applications in the paper. The ultimate goal is for me to  understand some of the more recent rigidity results for genera.

1. The Atiyah-Bott fixed point formula

Let be a compact manifold, and suppose given an endomorphism with finitely many fixed points. The classical Lefschetz fixed-point formula counts the number of fixed points via the supertrace of the action of on cohomology . In other words, if is the fixed point set, we have

where is a sign related to the determinant of at .

Using the de Rham isomorphism, the groups are identified with the cohomology of a complex of sections of bundles

This is an example of an elliptic complex of differential operators: in other words, when one takes the symbol sequence at a nonzero cotangent vector, the induced map of vector spaces is exact. It is a consequence of this that the cohomology groups are finite-dimensional.

The Atiyah-Bott fixed point formula is a striking generalization of the previous fact. Consider an elliptic complex of differential operators on ,

where the are vector bundles over . The cohomology groups of this complex are finite-dimensional and provide a generalization (not much of a generalization, actually) of the index of an elliptic operator; they thus often hold significant geometric information about .

 

The Hurewicz map and the logarithm of complex cobordism

In earlier posts, we analyzed the Hurewicz homomorphism

in purely algebraic terms: was the Lazard ring, and the Hurewicz map was the map classifying the formal group law for the “change of coordinates”

We can also express the Hurewicz map in more geometric terms. The ring is, by the Thom-Pontryagin construction, the cobordism ring of stably almost-complex manifolds. The ring is isomorphic to the Pontryagin ring by the Thom isomorphism. In these terms, we can describe the Hurewicz map explictly:

Theorem 1 The Hurewicz map sends the cobordism class of a stably almost complex manifold to the element where

classifies the stable normal bundle of (together with its complex structure), and where is the fundamental class of .

In particular, we will be able to work out explicitly where a given complex manifold representing a cobordism class goes under this map. As an application, we’ll show using Lagrange inversion that the complex projective spaces determine the logarithm of the formal group law on .

 

The formal group law of unoriented cobordism

Before moving on, I’d like to work out the analog for real-oriented cohomology theories (which is tangential to the rest of the story, though). This is considerably less interesting, but perhaps it’s a toy example of the ideas explained in the last few posts without the full-blown machinery of the Adams spectral sequence and so forth.

So, let’s state what the analogous ideas are in the real-oriented context:

1. A real-oriented ring spectrum  is a ring spectrum together with a functorial, multiplicative choice of Thom classes for real vector bundles; equivalently, there is a morphism of ring spectra

   

   That is, the universal real-oriented ring spectrum is unoriented cobordism, whose homotopy groups can be completely computed.

2. If is real-oriented, then is a -vector space, and all the usual computations of and so forth work just fine for , and there is a theory of Stiefel-Whitney classes in -cohomology.
3. Given a real-oriented spectrum , we thus have where is the Stiefel-Whitney class of the tautological bundle. Similiarly, .

Since is the classifying space for real line bundles, there is a monoidal product

classifying the tensor product of line bundles. As before, this means that we can extract a formal group law over . This formal group law has the property that if are two line bundles over a finite-dimensional space , then in ,

So far everything has been analogous to the complex-oriented case, but there is an extra feature here which changes the picture drastically. Namely, when one works with realline bundles , we have that is always trivial. (The analog is very false for complex line bundles.) This means that the formal group law must satisfy

This is not satisfied by, say, the multiplicative formal group law. (And in fact, -theory—the natural candidate for this—is not real-oriented, only spin-oriented.)

With this in mind, the analog of Quillen’s theorem for becomes:

Theorem 1 (Quillen) The formal group law for is the universal formal group law over a satisfying .

 

Quillen’s theorem on the formal group law of MU

Our goal is now to return to topology, and in particular to study the formal group law of the universal complex-oriented theory (complex cobordism). As we computed using the Adams spectral sequence,

This is the Lazard ring, by the computations of the previous couple of posts. On the other hand, it is not at all clear that the map

classifying the formal group law over (arising from the complex orientation) is actually an isomorphism: in other words, that the formal group law of is the universal one. The fact that it is in fact an isomorphism is the content of Quillen’s theorem, which will be proved in this post.

 

Finishing the proof of Lazard’s theorem

This is the third in the series of posts intended to work through the proof of Lazard’s theorem, that the Lazard ring classifying the universal formal group law is actually a polynomial ring on a countable set of generators. In the first post, we reduced the result to an elementary but tricky “symmetric 2-cocycle lemma.” In the previous post, we proved most of the symmetric 2-cocycle lemma, except in characteristic zero. The case of characteristic zero is not harder than the cases we handled (it’s easier), but in this post we’ll complete the proof of that case by exhibiting a very direct construction of logarithms in characteristic zero. Next, I’ll describe an application in Lazard’s original paper, on “approximate” formal group laws.

After this, I’m going to try to move back to topology, and describe the proof of Quillen’s theorem on the formal group law of complex cobordism. The purely algebraic calculations of the past couple of posts will be necessary, though.

1. Formal group laws in characteristic zero

The last step missing in the proof of Lazard’s theorem was the claim that the map

classifying the formal group law obtained from the additive one by “change-of-coordinates” by the exponential series is an isomorphism mod torsion. In other words, we have an isomorphism

In fact, we didn’t really need this: we could have proved the homological 2-cocycle lemma in all cases, instead of just the finite field case, and it would have been easier. But I’d like to emphasize that the result is really something elementary here. In fact, what it is saying is that to give a formal group law over a -algebra is equivalent to giving a choice of series .

Definition 1 An exponential for a formal group law is a power series such that

The inverse power series is called the logarithm.

That is, a logarithm is an isomorphism of with the additive formal group law.

So another way of phrasing this result is that:

Proposition 2 A formal group law over a -algebra has a unique logarithm (i.e., is uniquely isomorphic to the additive one).

 

Lazard’s theorem II

We are in the middle of proving an important result of Lazard:

Theorem 1 The Lazard ring over which the universal formal group law is defined is a polynomial ring in variables of degree .

The fact that the Lazard ring is polynomial implies a number of results which are not a priori obvious: for instance, it shows that given a surjection of rings , then any formal group law on can be lifted to one over .

We began the proof of Lazard’s theorem last time: we produced a map

classifying the formal group law obtained from the additive one by the “change of coordinates” . We claimed that the map on indecomposables was injective, and that, in fact the image in the indecomposables of could be determined completely. I won’t get into the details of this (it was all in the previous post), because the purpose of this post is to prove a result to which we reduced last time.

Let be an abelian group. A symmetric 2-cocycle is a “polynomial” with the properties:

and

These symmetric 2-cocycles come up when one tries to classify formal group laws over the ring , as we saw last time: in fact, we can think of them as “deformations” of the additive formal group law.

The main lemma which we stated last time was the following:

Theorem 2 (Symmetric 2-cocycle lemma) A homogeneous symmetric 2-cocycle of degree is a multiple of where if is not a power of a prime, and if .

For a direct combinatorial proof of this theorem, see Lurie’s notes. I want to describe a longer homological proof, which is apparently due to Mike Hopkins and which appears in the COCTALOS notes. The strategy is to interpret these symmetric 2-cocycles as actual cocycles in a cobar complex computing an group. Then, the strategy is to compute this group independently.

This argument is somewhat longer than the combinatorial one, but it has the benefit (for me) of engaging with some homological algebra (which I need to learn more about), as well as potentially generalizing in other directions.