Oriented cobordism III: Wall’s theorem

This is the third in a series of posts on oriented cobordism. In the first post, we analyzed the spectrum at odd primes; in this post, we will analyze the prime . After this, we’ll be able to deduce various classical geometric facts about manifolds.

The next goal is to  determine the structure of the homology as a comodule over . Alternatively, we can determine the structure of the cohomology  over the Steenrod algebra : this is a coalgebra and a module.

Theorem 8 (Wall) As a graded -module, is a direct sum of shifts of copies of and .

This corresponds, in fact, to a splitting at the prime 2 of into a wedge of Eilenberg-MacLane spectra.

In fact, this will follow from the comodule structure theorem of the previous post once we can show that if is the Thom class, then the action of on has kernel : that is, the only way a cohomology operation can annihilate if it is a product of something with . Alternatively, we have to show that the complementary Serre-Cartan monomials in applied to ,

are linearly independent in .

 

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.

 

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.