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.

**1. Bott’s integrality theorem**

To start with, I would like to mention the following theorem of Bott:

Theorem 2The top Chern class of any (complex) vector bundle on is divisible by .

To prove this, one shows that the Chern *character* of any complex vector bundle on is integral (though a priori it is only in the*rational* cohomology ring). This is done by introducing the reduced -group of stable classes of vector bundles; the Chern character defines a map

This map is additive and multiplicative. To check that the Chern character of *any* complex vector bundle on is integral, one thus reduces to analyzing the image of as above.

This is convenient because the group is very simple. By the Bott periodicity theorem, it is generated by the th power of where is the Hopf bundle over , so that

So, all we need to do is to check that in is integral; then, because of the multiplicative properties of the Chern character, the general result will follow.

However, is clearly an integer. Thus, the claim about is proved.

The final step is to translate from the statement about to a statement about . Here, one has to recall exactly how is defined. If is a vector bundle such that

then

If a subscript indicates a graded component, then this implies that

We note that the elementary symmetric polynomials are the Chern classes , and that there is a certain universal expression such that for any with the elementary symmetric polynomials as before,

We only need the first symmetric polynomials. Thus, we have that (where denotes the th component)

This is true for any vector bundle , not necessarily a sum of line bundles, by the splitting principle; actually, it is probably how one defines in the first place.

In any event, what we have proved is that

for any vector bundle on . In this case, of course, , so we only need the coefficient of in to conclude. Here we can use Newton’s identities to conclude that

This implies the integrality theorem, since we have seen that the left side is integral.

In fact, it seems that this integrality result can be deduced from the Atiyah-Singer index theorem by intepreting as the index of a “twisted signature operator”; I don’t understand this well enough to comment yet.

There are a lot of surprising integrality results on the characteristic classes of the *tangent bundle* to a manifold; for instance, the Hirzebruch signature formula implies that of the tangent bundle is always an integer for an oriented four-manifold. Another example is that the Todd genus of a complex manifold is always an integer, by the Hirzebruch-Riemann-Roch theorem. However, there don’t seem to be that many that apply to *every* vector bundle on a space.

**2. Almost complex structures on spheres**

Recall that an *almost complex structure* on a manifold is a complex structure on its tangent bundle. Any complex manifold is an almost complex manifold. Among the spheres, it is a classical theorem that only and admit almost complex structures.

We can deduce most of this result very quickly from Bott’s integrality theorem above.

Theorem 3No sphere other than and can admit an almost complex structure.

In fact, suppose the tangent bundle could be given the structure of a complex vector bundle. Then it would have Chern classes, and in particular, it would have a top Chern class . This is necessarily the Euler class , since the Euler class is the Euler characteristic. However, we have also seen that is divisible by , and for this cannot thus be (or rather, times a generator of ).

The argument does not show that does not admit an almost complex structure. But one can give a direct argument as follows. Suppose admitted the structure of a complex vector bundle. We know that since is stably trivial. However, if we assume that is a complex vector bundle then we have the following identity for the total Pontryagin class:

Necessarily is nonzero (as it is the Euler class, which is twice the generator of ). So this is a contradiction.

In fact, using the signature theorem, one can moreover conclude that any compact four-manifold with and nonzero Euler characteristic does not admit an almost complex structure. The argument is the same, once one notes that , at least after tensoring with .

**3. Atiyah’s theorem**

Atiyah’s theorem will be proved in a similar way as Bott’s integrality theorem, except that we will get something identically zero, since we are working in a torsion ring. The necessary tool is the (harder) form of Bott periodicity for -theory.

Namely, the periodicity theorem states that:

Theorem 4There is an eight-dimensional bundle over , whose Stiefel-Whitney in is nonzero, such that multiplication by determines an isomorphism

for any CW complex .

Recall that our goal is to show that the Stiefel-Whitney classes of any vector bundle over a nine-fold suspension vanish. To do so, we’ll use a relation between the Stiefel-Whitney classes of an element and those of . This is the analog of the calculation of the Chern classes of the generator of .

Let be the nonzero element. There is an isomorphism given by multiplication by .

Lemma 5If is a CW complex and , then

where is the polynomial which expresses the sum in terms of the elementary symmetric functions of .

Let us prove this. In fact, can be represented by for a real vector bundle of dimension over . Then is represented by

(modulo stable equivalence). So we need to compute the Stiefel-Whitney classes of this. As usual, we may pretend that is a sum of line bundles, with Stiefel-Whitney classes . We can also pretend that is a sum of line bundles with Stiefel-Whitney classes ; although this is not literally true any manipulations we make will be symmetric in the and thus will be justified by the splitting principle.

So,

In particular, since , and sends tensor products of line bundles to sums of cohomology classes, we also have

However, since all but one of the elementary symmetric functions of is zero, and their product is , we find:

Next, the Stiefel-Whitney classes of are

Finally, we have

If we put all these together, and use the fact that (in characteristic 2!), we find that

Now, again we can simplify if we note that . We can combine the two products to get another expression for :

Here because everything simplifies to

which gives the expression we desired, since the symmetric polynomials of the are the Stiefel-Whitney classes of .

**4. Proof of Atiyah’s theorem**

With the formula for established as in the lemma, it will be fairly quick to prove Atiyah’s theorem. The idea is the following. Consider an element . Then in particular, there is such that ; this means that

Here the live instead ; the advantage, however, is that this ring is very simple: all cup-products in a suspension vanish. Since the top term of is , and all the other terms involve nontrivial products (again, by Newton’s identities), we find that is , since we are in characteristic two.

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