The cohomology of projective hypersurfaces

Consider a smooth surface of degree . We are interested in determining its cohomology.

1. A fibration argument

A key observation is that all such ‘s are diffeomorphic. (When is replaced by , then this is just the observation that the genus is determined by the degree, in the case of a plane curve.) In fact, consider the space of all degree homogeneous equations, so that is the space of all smooth surfaces of degree . There is a universal hypersurface consisting of pairs where is a point lying on the hypersurface . This admits a map

which is (at least intuitively) a fiber bundle over the locus of smooth hypersurfaces. Consequently, if corresponds to smooth hypersurfaces, we get an honest fiber bundle

But is connected, since we have thrown away a complex codimension subset to get from ; this means that the fibers are all diffeomorphic.

This argument fails when one considers only the real points of a variety, because a codimension one subset of a real variety may disconnect the variety.

 

Bott periodicity and integrality theorems

Today I would like to blog about a result of Atiyah from the 1950s, from his paper “Bott periodicity and the parallelizability of the spheres.” Namely:

Theorem 1 (Atiyah) On a nine-fold suspension of a finite complex, the Stiefel-Whitney classes of any real vector bundle vanish.

In particular, this means that any real vector bundle on a sphere cannot be distinguished using Stiefel-Whitney classes from the trivial bundle. The argument relies on the Bott periodicity theorem and some calculations with Stiefel-Whitney classes. There is also an analog for the Chern classes of complex vector bundles on spheres; they don’t necessarily vanish but are highly divisible.

These sorts of integrality theorems often have surprising geometric consequences. In this post, I’ll discuss the classical problem of when spheres admit almost-complex structures, a problem one can solve using the second of the integrality theorems mentioned above. Atiyah was originally motivated by the question of parallelizability of the spheres.

 

Wu’s theorem on the Stiefel-Whitney classes of a manifold

Today I would like to take a break from the index theorem, and blog about a result of Wu, that the Stiefel-Whitney classes of a compact manifold (i.e. those of the tangent bundle) are homotopy invariant. It is not even a priori obvious that the Stiefel-Whitney classes are homeomorphism invariant; note that “homeomorphic” is a strictly weaker relation than “diffeomorphic” for compact manifolds, a result first due to Milnor. But in fact the argument shows even that the Stiefel-Whitney classes (of the tangent bundle) can be worked out solely in terms of the structure of the cohomology ring as a module over the Steenrod algebra.

Here is the idea. When is a closed submanifold of a manifold, there is a lower shriek (Gysin) homomorphism from the cohomology of to that of ; this is Poincaré dual to the restriction map in the other direction. We will see that the “fundamental class” of (that is,  the image of 1 under this lower shriek map) corresponds to the mod 2 Euler (or top Stiefel-Whitney) class of the normal bundle. In the case of , the corresponding normal bundle is just the tangent bundle of . But by other means we’ll be able to work out the Gysin map easily. Once we have this, the Steenrod operations determine the rest of the Stiefel-Whitney classes.

 

Thom’s construction of the Stiefel-Whitney classes

I’ve been trying to fix the (many) gaps in my knowledge of classical algebraic topology as of late, and will probably do a few posts in the near future on vector bundles, K-theory, and characteristic classes.

Let be a base space, and let be a real vector bundle. There are numerous constructions for the characteristic classes of . Recall that these are elements in the cohomology ring (for some ring) that measure, in some sense, the twisting or nontriviality of the bundle .

Over a smooth manifold , with a smooth vector bundle, a construction can be made in de Rham cohomology. Namely, one chooses a connection on , computes the curvature tensor of (which is an -valued 2-form on ), and then applies a suitable polynomial from matrices to polynomials to the curvature . One can show that this gives closed forms, whose de Rham cohomology class does not depend on the choice of connection. This is the subject of Chern-Weil theory, and it applies more generally to principal -bundles on a manifold for a Lie group.

But there is something that this approach misses: torsion. By working with de Rham cohomology (or equivalently, cohomology with -coefficients), the very interesting torsion phenomena that algebraic topologists care about is lost. For the purposes of this post, we’re interested in cohomology classes where the ground ring is , and so de Rham cohomology is out. However, in return, we have cohomology operations. We can use them instead.

 

Chern-Weil II

I’d like to finish the series I started a while back on Chern-Weil theory (and then get back to exponential sums).

So, in the discussion of the Cartan formalism a few days back, we showed that given a vector bundle with a connection on a smooth manifold, we can associate with it a curvature form, which is an -valued 2-form; this is a generalization of the Riemann curvature tensor (as some computations that I don’t feel like posting here will show). In the case of a line bundle, we saw that since was canonically trivialized, we could interpret the curvature form as a plain old 2-form, and in fact it turned out to be a representative — in de Rham cohomology — of the first Chern class of the line bundle. Now we want to see what to do for a vector bundle, where there are going to be a whole bunch of Chern classes.

For a general vector bundle, the curvature (of a connection) will not in itself be a form, but rather a differential form with coefficients in , which is generally not a trivial bundle. In order to get a differential form from this, we shall have to apply an invariant polynomial. In this post, I’ll describe the proof that one indeed gets well-defined characteristic classes (that are actually independent of the connection), and that they coincide with the usually defined topological Chern classes.

 

Chern-Weil for complex line bundles

So, now with the preliminaries on connections and curvature established, and the Chern classes summarized, it’s time to see how they connect with one another. Namely, we want to say that, given a complex vector bundle, we can compute the Chern classes in de Rham cohomology by picking a connection — any connection — on it,  computing the curvature, and then applying various polynomials.

We shall start by warming up with a special case, of a line bundle, where the algebra needed is easier. Let be a smooth manifold, a complex line bundle. Let be a connection on , and let be the curvature.

Thus, is a global section of ; but since is a line bundle, this bundle is canonically identified with . (Recall the notation that is the bundle (or sheaf) of smooth -forms on the manifold .)

Proposition 1 (Chern-Weil for line bundles)  is a closed form, and the image in is times the first Chern class of the line bundle .

 

Chern classes

So, I’m in a tutorial this summer, planning to write my final paper on the Kodaira embedding theorem, and I’ve been finding my total ignorance of complex algebraic geometry to be something of a problem. One of my goals next year is, coincidentally, to acquire a solid understanding of most of the topics in Griffiths-Harris. To start with, I’d like to spend a few posts on Chern-Weil theory. This gives an analytic method of computing the Chern classes of a complex vector bundle, and more generally a framework for the characteristic classes of a principal bundle over a Lie group. In fact, it tells you what the cohomology of the classifying space of a Lie group is (it’s a certain algebra of invariant polynomials on the Lie algebra), from which — by Yoneda’s lemma — you can associate cohomology classes to a principal bundle on any space.

Today, I’d like to review what Chern classes are like.

1. Introduction

To start with, we will need to describe what the Chern classes really are. These are going to be natural maps

from the complex vector bundles on a space to the cohomology ring. In other words, to each vector bundle , we will have an element . In order for this to be natural, we are going to want that, for any map of topological spaces,

In other words, we are going to want the map to be functorial in , when both are considered as contravariant functors in . It turns out that each functor (of -dimensional complex vector bundles) and is representable on the appropriate homotopy category.

 

The Stiefel-Whitney classes of projective space

Last time we gave the axiomatic description of the Stiefel-Whitney classes. Today, following Milnor-Stasheff, we want to look at what happens in the particular case of real projective space . In particular, we want to compute the Stiefel-Whitney classes of the tangent bundle . The cohomology ring of with -coefficients is very nice: it’s . We’d like to find what is.

On , we have a tautological line bundle such that the fiber over is the set of vectors that lie in the line represented by . Let’s start by figuring out the Stiefel-Whitney classes of this. I claim that

The reason is that, if is a linear embedding, then pulls back to the tautological line bundle on . In particular, by the axioms, we know that , and in particular has nonzero . This means that by the naturality. As a result, is forced to be , and there can be nothing in other dimensions since we are working with a 1-dimensional bundle. The claim is thus proved.

 

Stiefel-Whitney classes

The first basic example of characteristic classes are the Stiefel-Whitney classes. Given a (real) -dimensional vector bundle , the Stiefel-Whitney classes take values in the cohomology ring . They can be used to show that most projective spaces are not parallelizable.

So how do we get them? One way, as discussed last time, is to compute the cohomology of the infinite Grassmannian. This is possible by using an explicit cell decomposition into Schubert varieties. On the other hand, it seems more elegant to give the axiomatic formulation. That is, following Milnor-Stasheff, we’re just going to list a bunch of properties that we want the Stiefel-Whitney classes to have.

Let be a bundle. The Stiefel-Whitney classes are characteristic classes that satisfy the following properties.

First, when . When you compute the cohomology of , the result is in fact a polynomial ring with generators. Consequently, we should only have characteristic classes of an -dimensional vector bundle. In addition, we require that always.

Second, like any characteristic class, the are natural: they commute with pulling back. If is a bundle, is a map, then . Without this, they would not be very interesting.

 

Vector bundles, Grassmannians, and characteristic classes

I’ve been reading about spectra and stable homotopy theory lately, but don’t feel ready to start talking about them here. Instead, I shall say a few words on characteristic classes. The present post will be quite general and preparatory — the more difficult matter is to actually construct such characteristic classes. Our goal is to see that characteristic classes essentially boil down to computing the cohomology of the infinite Grassmannian.

A lot of problems in mathematics involve the existence of sections to vector bundles. For instance, there is the old question of when the sphere is parallelizable. A quick Euler characteristic argument shows that even-dimensional spheres can’t be—then there would be an everywhere nonzero vector field, whose infinitesimal flows would be homotopic to the identity (and consequently having nonzero Lefschetz number by the even-dimensionality) while having no fixed points. In fact, much more is known. Using the group or group-like structures on (coming from the complex numbers, quarternions, and octonions), it is easy to see that these manifolds are parallelizable. But in fact no other sphere is.

A characteristic class is a means of assigning some invariant to a vector bundle. Ideally, it should be trivial on trivial bundles, so the characteristic class can be thought of as an “obstruction” to finding large numbers of linearly independent sections.

More formally, let be a vector bundle. A characteristic class assigns to this bundle (of some fixed dimension, say ) an element of the cohomology ring (with coefficients in some ring). To be interesting, the characteristic class has to be natural. That is, if is a map, then the characteristic class of the pull-back bundle should be the pull-back of the characteristic class of .