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 ${Y = \Sigma^9 X}$ 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 $S^n, n \geq 9$ 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. (more…)

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 ${G}$ be a compact Lie group, as always. Let ${(X, A)}$ be a pair of compact ${G}$-spaces (with ${ A \subset X}$); then we have defined therelative ${K_G}$-theory via $\displaystyle K_G(X, A) \equiv \widetilde{K}_G(X/A).$

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

To describe ${K_G(X, A)}$ without use of the space ${X/A}$, we can proceed as follows.

Definition 1 We let ${C_G(X, A)}$ be the category of complexes of ${G}$-vector bundles $\displaystyle 0 \rightarrow E_0 \rightarrow \dots \rightarrow E_n \rightarrow 0$

on ${X}$, which when restricted to ${A}$ are exact. A morphism in ${C_G(X, A)}$ is a map of chain complexes.

The idea is that we are going to assign to every element ${E_\bullet}$ of ${C_G(X,A)}$ an element of the relative K-theory ${K_G(X, A) = \widetilde{K}_G(X/A)}$, by effectively taking the alternating sum ${\sum (-1)^i [E_i]}$. In order to do this, we will start by modifying the complex ${E_\bullet}$ by adding acyclic complexes. Namely, we start by adding complexes of the form $\displaystyle 0 \rightarrow 0 \rightarrow \dots \rightarrow 0 \rightarrow F \rightarrow F \rightarrow 0 \rightarrow 0 \rightarrow \dots$

where ${F}$ is a ${G}$-vector bundle on ${A}$, to make all but the first term of ${E_\bullet}$ trivial (i.e. coming from an ${R(G)}$-representation). With this change made, we can assume that all but the first term of ${E_\bullet}$ is stably trivial. Then the first term of ${E_\bullet}$ is stably trivial when restricted to ${A}$ by exactness of ${E_\bullet|_A}$. Consequently, we can quotient all the terms by ${A}$ and get a complex of ${G}$-vector bundles ${E_\bullet|_{X/A}}$; this is exact at the basepoint of ${X/A}$. Now, taking the alternating sum as desired, we get a map $\displaystyle C_G(X, A) \rightarrow K_G(X, A).$

This map does not see stable equivalence; that is, if we add to a complex ${E_\bullet}$ a complex of the form ${0 \rightarrow F \rightarrow F \rightarrow 0}$, the image in ${K_G(X, A)}$ does not change. Moreover, it is homotopy invariant.

In fact, one can give a presentation of the group ${K_G(X, A)}$ in this way. We start with the set of all such complexes in ${C_G(X, A)}$ 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 ${E_\bullet, F_\bullet \in C_G(X, A)}$ which can be obtained by restriction to the end faces of a complex in ${C_G(X \times [0, 1], A \times [0, 1])}$, then they should both be identified. Given these identifications, one gets precisely the group ${K_G(X, A)}$.

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. (more…)