Chromatic homotopy theory I

Let be a multiplicative cohomology theory. We say that is complex-oriented if one is given the data of an element which restricts to the canonical generator of . It turns out that one has a bit more: a complex orientation gives on a functorial, multiplicative choice of Thom classes for complex vector bundles. In fact, this is a perhaps more natural definition of such a theory.

What does this mean? Given a vector bundle , one can form the Thom space : in other words, the quotient of the unit ball bundle in (with respect to a choice of metric) by the unit sphere bundle . When is compact, this is just the one-point compactification of .

Definition 1 The vector bundle is orientable for a multiplicative cohomology theory if there exists an element which restricts to a generator on each fiberwise , where . Such a is called a Thom class.

Observe that for each point , there is a restriction map if the dimension of is .

The existence of a Thom class implies a Thom isomorphism, as for ordinary homology.

Theorem 2 (Thom isomorphism) A Thom class induces an isomorphism

given by cup-product with .

In the case of ordinary homology, a Thom class is unique (up to sign) if it exists; in general, though, a Thom class is highly non-unique, and an orientation is additional data than simple orientability.

Here are a few basic cases:

1. Any vector bundle is orientable for -cohomology.
2. An oriented (in the usual sense: i.e., the top wedge power is trivial) vector bundle is one oriented for -cohomology.
3. Complex vector bundles are oriented for -theory. We will see this below.
4. Spin bundles are oriented for -theory. An explicit construction of Thom classes can be made, as virtual bundles arising from Clifford modules: this is in Atiyah-Bott-Shapiro’s paper.
5. A trivial bundle is orientable for any cohomology theory (this is rather uninteresting: the Thom space is just a suspension).

 

Equivariant K-theory III

Next, I would like to describe an alternative description of relative K-theory which is sometimes convenient (e.g. when describing the Thom isomorphism in K-theory). Let be a compact Lie group, as always. Let be a pair of compact -spaces (with ); then we have defined therelative -theory via

Here is equipped with a distinguished basepoint (corresponding to ), and as a result this makes sense. As usual, we can use this definition to make into a cohomology theory on compact -spaces.

To describe without use of the space , we can proceed as follows.

Definition 1 We let be the category of complexes of -vector bundles

on , which when restricted to are exact. A morphism in is a map of chain complexes.

The idea is that we are going to assign to every element of an element of the relative K-theory , by effectively taking the alternating sum . In order to do this, we will start by modifying the complex by adding acyclic complexes. Namely, we start by adding complexes of the form

where is a -vector bundle on , to make all but the first term of trivial (i.e. coming from an -representation). With this change made, we can assume that all but the first term of is stably trivial. Then the first term of is stably trivial when restricted to by exactness of . Consequently, we can quotient all the terms by and get a complex of -vector bundles ; this is exact at the basepoint of . Now, taking the alternating sum as desired, we get a map

This map does not see stable equivalence; that is, if we add to a complex a complex of the form , the image in does not change. Moreover, it is homotopy invariant.

In fact, one can give a presentation of the group in this way. We start with the set of all such complexes in as above. We identify complexes which are chain homotopic to each other. Then, we mod out by the relation of (geometric) homotopy: if one has complexes which can be obtained by restriction to the end faces of a complex in , then they should both be identified. Given these identifications, one gets precisely the group .

I don’t really want to prove these things in detail, partially because I don’t want to get too bogged down with this project.

 

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 Thom isomorphism theorem

Suppose is a topological space, and is a vector bundle over . Here is a “reasonable” space, for instance a manifold or a CW complex. There are many familiar examples of how such bundles arise “in nature,” for instance the tangent bundle to a manifold. It is of interest to understand how the apparatus of algebraic topology applies to the bundle.

Now, clearly there is a deformation retraction of given by that deformation retracts to the image of the zero section. In particular, is homotopy equivalent to . So by itself might not all be that interesting.

Nonetheless, let us suppose that is a normed bundle, e.g. the tangent bundle to a manifold with a Riemannian metric. In this case, we can consider the set of elements of norm one; it is no longer a vector bundle over , but it is a fiber bundle (with fibers various spheres). This is homotopically very different from in general.

For a general bundle , we can still consider the subset consisting of nonzero elements. Then there is a map whose fibers are of the form . It is easy to see that if is a normed bundle, then deformation retracts onto the associated sphere bundle. So we might as well see what is like. By using the long exact sequence, we might try studying the pair .

Today, I want to talk about the cohomology of the pair . Namely, the main result is:

Theorem 1 (Thom) For all , there is an isomorphism (to be described)

if has dimension .

(To ease notation, all cohomology will henceforth be with -coefficients.)