A category is concrete if there is a faithful functor . Most of the categories one initially encounters are in fact concrete: categories of groups, rings, modules, Lie algebras, and so on, and one can think of them as consisting of “structured sets” and “morphisms respecting that structure.” Every small category is concrete, because one can take the Yoneda embedding

followed by the product functor .

Nonetheless, not every category is concrete, and the following example shows that a very natural one is not:

Theorem 1 (Freyd) The homotopy category of pointed spaces is not concrete.

In other words, a homotopy type is somehow too complex to be encoded simply as a set with appropriate structure.

The idea of the proof is essentially the following. In a category of structured sets, a given object can only have so many subobjects, because a set has only so many subsets. But there are categories where an object may have an enormous collection of subobjects, because the definition of a subobject is purely arrow-theoretic. So a category where objects can have lots of subobjects is probably not concrete.

1. The categorical input

Let me now try to set down the argument in a bit more detail. The homotopy category has a special property: it is a pointed category, that is, the initial and terminal objects are the same, and called the zero object. This means that for any two objects , there is a canonical zero morphism  (which can be realized by the constant map to the basepoint). Freyd considers, instead of faithful functors , faithful pointed functors

to the category of pointed sets. A pointed functor is one preserving the zero object (and hence the zero morphisms). The following observation becomes useful:

Observation: If a pointed category is concrete, then there is a faithful pointed functor .

The proof is to start with a functor (not necessarily pointed), apply the free abelian group functor, and quotient by the image of the zero object.

The advantage of working in purely a pointed setting is that one can talk about the kernel of a map in a pointed category: it is the equalizer of with the zero morphism.

Next, we come to the set-theoretic business. It is somewhat awkward to deal with subobjects in the homotopy category, so Freyd considers something a bit more general: given an object , he considers an equivalence relation on all morphisms (rather than simply monomorphisms).

Definition 2 Let be a pointed category. We define an equivalence relation on the class of maps for a fixed object . We say that and are equivalent if, for any map , then either and are zero or neither is zero.

This use of equivalence relations on proper classes seems a bit theological, but essentially all we’re going to be interested in the following is whether the set of equivalence classes forms a set.

Definition 3 An object is good if there is a set of equivalence classes of morphisms .

For instance, every object in is good.

One can, of course, phrase this definition in a manner that makes no appeal to proper classes: is good if one has a set of morphisms such that any morphism is equivalent (in the above sense) to one of them.

Here is the main concretizability criterion in the paper.

Theorem 4 (Freyd) Let be a pointed category which admits a pointed functor sending nonzero morphisms to nonzero morphisms (e.g. a concrete category). Then every object of is good.

In fact, Freyd proves the converse as well, but we don’t need it.

The proof of this result is now straightforward. Let be an object of , and let be two morphisms. Then if is not equivalent to , it follows that is not equivalent to . This is because sends nonzero morphisms to nonzero morphisms. This means that there can’t be a huge amount of equivalence classes of morphisms because there aren’t that many equivalence classes of morphisms of pointed sets .

Alternatively, we could phrase this as follows:

Observation: A pointed functor sending nonzero morphisms to nonzero morphisms reflects goodness.

So now, since every object of is good, it follows that every object of must be good. This completes the proof.

To prove that the homotopy category is not concretizable, we’ll now need to produce un-good objects of . In fact, Freyd shows that the suspension of a Moore space is such an example. In order to get this, we’ll need to describe a general method of showing that an object is un-good.

2. Producing un-good objects

A simple example of producing morphisms for a fixed object (a fixed pointed category) is to take maps out of and then to take their kernels. This isn’t very easy to do in the homotopy category; however, we do have something slightly weaker that turns out to be useful.

Definition 5 Let be a morphism in . A weak kernel of is a map

such that for any object , we have an exact sequence of pointed sets

This means that is zero, and conversely, is “versal” with respect to this property. That is, any map which followed by is zero factors non-uniquely through . An example is provided by the theory of fiber sequences in homotopy theory.

We note the following:

Proposition 6 Two maps which are weak kernels (possibly of different maps) are equivalent if and only if factors through and factors through .

The proof follows from the definitions.

3. Dualization

In the context of Freyd’s theorem, it is useful, however, to dualize everything: that is, to work not in but in the opposite category. It is easy to check that a category is concrete if and only if its opposite category is concrete. Instead of using the language of the opposite category, we might as well just dualize everything said before. That is, in a pointed category , for an object , there is a natural equivalence relation on maps . We can thus define an object to be co-good if The earlier theorem still applies and shows that if admits a pointed, zero-reflecting functor to , then must possess only co-good objects.

So, how might we produce co-bad (I’m going to resist the temptation to say co-un-good) objects of ? As in the previous section, a natural thing to do is to consider maps

which are weak cokernels (defined dually as in weak kernels) to some map into . Two weak cokernels are equivalent if and only if they factor through each other.

However, there is a very large supply of weak cokernels in homotopy theory: by the Barratt-Puppe sequence, the suspension of any map is a weak cokernel. Freyd’s strategy is to produce a space (the Moore space) together with a lot of maps

for other -torsion groups (where is an arbitrary ordinal).

The suspensions of these maps will turn out not to factor through one another, and consequently one gets a class’s worth of non-equivalent maps out of . This is thus a co-bad object, which proves the non-concreteness of the homotopy category.

4. A little group theory

Surprisingly, the construction will make use of some theory of abelian groups. Let us work with -torsion abelian groups here, for some fixed prime. We will find useful the notion of the height of such a group (actually, that will be sort of implicit).

Definition 7 For an ordinal , we define the functor on the category of -torsion groups as follows by transfinite induction. 

1. If , then is just the usual definition: the subgroup of all multiples of in .
2. If is defined, we set to be . This defines the functor for non-limit ordinals.
3. If is a limit ordinal, then we define

   

These can be nonzero for ordinals much bigger than the countable one, as we’ll see.

For an ordinal , we can give an example of a -torsion group such that consists of a . This means that is zero when .

Construction: We let be defined by the following generators: we take all finite ascending sequences of ordinals with ; the corresponding generator is denoted . The empty sequence corresponds to zero.

We impose the relation that

This guarantees that every element is -torsion. The claim is that this group does it. In fact, one can see by transfinite induction that is the subgroup generated by sequences with . So, is a on the generator .

Notation: We write for this particular generator.

Proposition 8 For , any homomorphism sends to zero.

In fact, that’s because , but . So this is just functoriality.

5. Freyd’s construction

Now we will be able to give Freyd’s construction. Let be an integer. Recall that for every ordinal , we constructed a group together with a map

coming from the generator . We saw that there was no way to find a commutative diagram

if . In other words, the ‘s can’t be factored through each other.

This suggests that if we realize the ‘s in terms of homology (or homotopy), then we’ll be able to solve the problem we desired in . So in fact, let’s consider the Moore spaces and ; we can define maps

inducing the required maps on homology. Then we get an infinite sequence of maps out of which cannot be factored through each other, because this factorization can’t even be done at the level of homology.

So we would be able to conclude that the maps were all non-equivalent to each other for different ‘s if they were weak cokernels. They might not be, but we can fix that: suspend. We can conclude that the maps are all pairwise non-equivalent. Thus is a co-bad space in , and the category cannot be concrete.

Incidentally, it’s possible to rewrite this proof using not homology but homotopy, and replacing the Moore spaces by Eilenberg-MacLane spaces; then one would have to take the loop space of a map rather than the suspension, and use the dual Barratt-Puppe sequence to argue about badness. However, Freyd wants to restrict attention to categories of finite-dimensional CW complexes and taking suspensions (unlike taking loops) preserves that.

Edit: It actually seems that WordPress has provided a means to do this directly. Let’s see how that works.

A classical problem (posed by Serre) was to determine whether there were any nontrivial algebraic vector bundles over affine space , for an algebraically closed field. In other words, it was to determine whether a finitely generated projective module over the ring is necessarily free. The topological analog, whether (topological) vector bundles on are trivial is easy because is contractible. The algebraic case is harder.

The problem was solved affirmatively by Quillen and Suslin. In this post, I would like to describe an elementary proof, due to Vaserstein, of the Quillen-Suslin theorem.

1. Stable freeness

An initial step, already taken by Serre, was to show that any finitely generated projective module over a polynomial ring (for a field) is stably free. Recall that a finitely generated module is said to be stably free if it becomes free after adding a finitely generated free module.

Remark: Given a projective module , there is always a free module such that is free. To see this, first choose aprojective module such that is free, and then take . It is easy to see that and that is free (if one appropriately groups the terms); this is the Eilenberg swindle. So, the finiteness conditions are really necessary here.

By the Serre-Swan theorem, one should think of projective modules as vector bundles, and, in particular, if is a compact Hausdorff space, we can actually identify (via an equivalence of categories) vector bundles on with finitely generated projective modules over the ring of continuous functions . Then, it follows that:

Proposition 1 A stably free module over is the same thing as a stably trivial vector bundle on : that is, a vector bundle that becomes trivial after adding a trivial vector bundle.

This observation allows one to get a simple example of a stably free module which is not free. The tangent bundle to is stably trivial (in fact, its one-dimensional normal bundle is trivial), but it is not trivial unless by a famous theorem (which is in fact a consequence of the Hopf invariant one theorem).

The first part of the proof of the Quillen-Suslin theorem is accomplished by:

Theorem 2 Let be a noetherian ring such that every finitely generated projective module over is stably free. Then the same property holds true for .

By induction, we see:

Corollary 3 Every finitely generated projective module over , for any field , is necessarily stably free.

This result is actually a special case of a theorem of Grothendieck. Given a ring (say, noetherian) , we can form the group , which is defined to be the Grothendieck group of the category of finitely generated projective -modules. Two projective modules map to the same element of if and only if there is a finite free module such that . Consequently, if and only if every projective -module is stably free.

The next result of Grothendieck is thus a generalization of the previous theorem:

Theorem 4 For a ring , extension of scalars induces an isomorphism .

The same is actually true of the higher K-groups, by a theorem of Quillen. I won’t describe the proof here.

2. Unimodular vectors

The main step is to go from “stably free” to free. Equivalently, we have to show that if we let , then any split injection

has a free cokernel. Let us start with the case ; this will turn out to be sufficient. We are interested in a condition such that any split injection will have a free cokernel, which is to say that is isomorphic to the canonical imbedding sending an element to .

We can reformulate the problem in a possibly more intuitive way. To give a split injection is the same as giving a vector whose components generate the unit ideal in . To say that the injection induced is isomorphic to the standard inclusion is to say that there is an isomorphism of taking to the vector . Alternatively, it is to say that the element can be completed to a basis for .

Definition 5 Let be any ring. A vector is unimodular if its components generate the unit ideal in . For two unimodular vectors , we write

if there is a matrix such that . This is clearly an equivalence relation.

So, the problem we are faced with now is to show that, for the rings of the form , any two unimodular vectors are equivalent. Alternatively, we have to check when one is equivalent to the standard one . Stated another way, we have to check whether there is an automorphism of carrying onto . If we can show this, then it will follow that any split injection has a free cokernel.

Here is an easy first step:

Proposition 6 Over a principal ideal domain , any two unimodular vectors are equivalent.

Proof: In fact, unimodular vectors correspond to imbeddings which are split injections. But if we have a split injection in this way, the cokernel is free (as we are over a PID), and consequently there is a basis for one of whose elements is . This implies that is conjugate to .

In a similar manner, if we use the fact that a finitely generated projective module over a local ring is free, then we obtain:

Corollary 7 Over a local ring , any two unimodular vectors are equivalent.

3. Polynomial rings over a local ring

The proof of the Quillen-Suslin theorem is essentially to induct on the number of variables. To do this, we’ll need an auxiliary result which states that, under mild hypotheses, a unimodular vector in a polynomial ring is equivalent to a unimodular vector in the base ring. This will be proved locally—one prime at a time. So, we start with:

Theorem 8 (Horrocks) Let for a local ring. Then any unimodular vector in one of whose elements has leading coefficient one is equivalent to .

Proof: Let be a unimodular vector. Suppose without loss of generality that the leading coefficient of is one, so that . If , then is a unit and there is nothing to prove. We will induct on .

Then, by making elementary row operations (which don’t change the equivalence class of ), we can assume that all have degree . Consider the coefficients of these elements. At least one of them must be a unit. In fact, if we reduce mod , then not all the can go to zero or the would not generate the unit ideal mod . So let us assume that contains a unit among its coefficients.

The claim is now that we can make elementary row operations so as to find another unimodular vector, in the same equivalence class, one of whose elements is monic of degree . If we can show this, then induction on will easily complete the proof.

Now, here is a lemma: If we have two polynomials , with and monic, and of degree containing at least one coefficient which is a unit, there is a polynomial of degree whose leading coefficient is one. This is easy to see with a bit of explicit manipulation.

This means that there are , such that has degree and leading coefficient a unit. If we keep this fact in mind, we can, using row and column operations, modify the vector such that it contains a monic element of degree . We just add appropriate multiples of to to make the leading coefficient a unit. This works if . If or , the lemma can be checked directly.

Consider the ring , and let be a unimodular vector. We want a condition to conclude that , where is the vector obtained by pointwise substitution. This will be the inductive argument we need for the Quillen-Suslin theorem. We already have a good criterion for when this is true in the case local.

Corollary 9 If is local and is a unimodular vector one of whose elements is monic, then .

In fact, is a unimodular vector in , hence equivalent to . We have also seen that is equivalent to .

The goal of the next step is to generalize this to the case where is not assumed local.

4. Localization

\newtheorem{lemma}{Lemma} Let be a domain. We start by observing that if in , then over . In fact, by hypothesis there is a matrix such that

which means that

We just have to then observe that

so we can take as the relevant matrix taking into .

The next lemma will be the required step to reduce to the case of local.

Lemma 10 Suppose over the localization . Then there exists a such that over .

Proof: As before, we can choose a matrix such that , and then the matrix has the property that

It follows that if we substitute for , then we have

The claim is that we can choose such that actually has -coefficients. In fact, this is because , which implies that for some matrix with values in . If we replace with for an element of , then we can clear the denominators in and arrange it so that .

Here, now, is the promised result which will be the crucial inductive step:

Corollary 11 Suppose is any ring, and is a unimodular vector one of whose leading coefficients is one. Then .

Proof: Let us consider the set of such that in . If we can show that , then we will be done, because after applying the homomorphism , we will get that in .

We start by observing that is an ideal. In fact, suppose and . Then, substituting in the first leads to

and since , we get easily by transitivity that . Similarly, we have to observe that if and , then . But this is true because one can substitute .

Since is an ideal, to show that we just need to show that is contained in no maximal ideal. Let be a maximal ideal. We then note that, by what we have already done for local rings, we have that

By the lemma, this means that there is a such that ; this means that . So cannot be contained in . Since this applies to any maximal ideal , it follows that must be the unit ideal.

5. The Quillen-Suslin theorem

With all these preliminaries, it will be relatively straightforward to establish the main result; the first step is to show that unimodular vectors over a polynomial ring are all equivalent.

Theorem 12 Let be a polynomial ring over a principal ideal domain , and let be a unimodular vector. Then .

Proof: We can now prove this by induction on . When , it is immediate.

Suppose . Then we can treat as , where we replace by to make it stand out. We can think of as a vector of polynomials in with coefficients in the smaller ring .

If has a term with leading coefficient one, then the previous results enable us to conclude that , and as lies in we can use induction to work downwards. The claim is that, possibly after a change of variables , we can always arrange it so that the leading coefficient in is one. The relevant change of variables leaves constant and

If is chosen very large, one makes by this substitution the leading term of each of the elements of a unit. So, without loss of generality we can assume that this is already the case. Thus, we can apply the inductive hypothesis on to complete the proof.

Theorem 13 (Quillen-Suslin) A finitely generated projective module over for a principal ideal domain is free.

In fact, we have to show that a stably free module over is free. That is, if is such a finitely generated module such that , then is free. By induction on , one reduces to the case . In this case we have an exact sequence

and we have to conclude that the cokernel is free.

But the injection corresponds to a unimodular vector, and we have seen that this is isomorphic to the standard embedding , whose cokernel is obviously free. Thus is free.

Earlier, I described an observation (due to Beck) that loop spaces could be characterized as algebras over the monad . At least, any loop space was necessarily an algebra over that monad, and conversely any algebra over that monad was homotopy equivalent to a loop space. There is an alternative and compelling idea of Segal which gives a condition somewhat easier to check.

As far as I understand, most of the different approaches to delooping a space consist of imitating the classical construction for a topological group : the construction of the space . It is known that any topological group is (weakly) homotopy equivalent , and conversely (though perhaps it is not as well known) that any loop space is homotopy equivalent to a topological group. (This can be proved using the simplicial construction of Kan.) Given a space (which may not be a topological group), the idea is that delooping machinery will assume given just enough structure to build something analogous to the classifying space, and then build that. This is, for instance, how the construction of Beck ran.

Here’s Segal’s idea; it is quite similar to the -idea. Given a topological group , we can construct using a standard simplicial construction. If is only a group object in the homotopy category, we can’t run this construction. Segal decides just to assume that one has given the data of a simplicial object that behaves like should and runs with that.

The starting point is that one can encode the structure of a monoid in a simplicial set. Given a monoid , the simplicial set has the following properties.

1. is a point.
2. The map induced by the inclusions (sending and to consecutive elements) is an isomorphism.

In fact, if we have any simplicial set with the above properties, it determines a unique monoid. This is proved in a similar way. If is such a simplicial set, then we take as the underlying set of the monoid, and the map comes from the boundary map ; the identity element comes from the map . So monoids can be described as simplicial sets satisfying certain properties (just as commutative monoids can).

As before, we can weaken this by replacing “isomorphism” by “homotopy equivalence.”

Definition 4 An S-datum is a simplicial space (e.g. bisimplicial set) with the following properties.

1. is (weakly) contractible.
2. The map induced by the inclusions (sending and to consecutive elements) is a weak equivalence.

The terminology should not be taken seriously because I just made it up.

As before, an S-datum determines an honest monoid object in the homotopy category, given by . The S-datum consists of a lifting of the associated simplicial object in the homotopy category to the category of topological spaces, and we will see that it allows one to build a classifying space. As we will see, a connected space which is not a loop space cannot admit an S-datum in this way. An incidental consequence is the existence of diagrams in the homotopy category of spaces which cannot be lifted to the category of spaces (choose a connected, homotopy associative H space which is not a loop space); see this MO question for simpler examples.

1. Segal’s result

The next result is the analog of the fact that .

Theorem 5 (Segal) If is an S-datum and is connected, then we have a weak equivalence

The proof of this relies on a bit of homotopy theory, of the formal and categorical flavor. The strategy is to work simplicially throughout, so is really a bisimplicial set—that is, a simplicial object in the category of simplicial sets. Then the “geometric realization” (which is itself a simplicial set) can be thought of in two ways: first of all, it is the “diagonal” simplicial set. If we think of as a family of sets depending on two parameters (because of the double occurrence of the word “simplicial”), then the diagonal simplicial set is . For another approach, it is the “homotopy colimit” (or higher categorical colimit) of the individual simplicial sets .

Now, the statement of Segal’s theorem is that there should be a homotopy cartesian diagram

where is really a stand-in for a contractible space. We will construct by taking the “simplicial path space.” Namely, we consider the map functor which sends , and to a simplicial object we take . There is an “extra degeneracy,” which means that the simplicial path space is always contractible, and its geometric realization must be too.

So, we’ve started with this simplicial space (space meaning “simplicial set”) , and we’ve formed the contractible simplicial space ; this has the property that

Thus we get a morphism of simplicial spaces (which is given by the last face map), such that the geometric realization of is (weakly) contractible. There is a commutative diagram

This comes from the natural imbedding of the zero-simplices in the geometric realization, at each stage. If we can prove that (1) is homotopy cartesian, then we’ll be done.

Now we are in the following situation. We have an indexing category , and two functors . There is a natural tranformation , which induces a natural transformation

Here can be given explicitly using the Bousfield-Kan formula. Then we have , and we want to show that the induced diagram

is homotopy cartesian.

2. A technical result

This distributivity property desired is certainly not true in general. However, there is a result that gives a criterion for this to happen.

Theorem 6 Consider two functors with the property that for each morphism in , there is a homotopy cartesian diagram

Then there are homotopy cartesian diagrams for each as in (2).

Let’s take this result as a black box (a nice reference is this paper of Charles Rezk), and use it to prove the theorem of Segal. We have a morphism of simplicial spaces (bisimplicial sets) , and we have to check the homotopy cartesian criterion of the above theorem to show that as claimed.

So, what we need to do is to show that for any map , the diagram

is homotopy cartesian.

Here is an example of what is going on. Let’s take , so we have the diagram

where the first horizontal map is, up to homotopy, the iterated multiplication and the first vertical map is, up to homotopy, multiplication of two of the factors . We have to check that the homotopy fibers are the same. The homotopy fiber on the right is clearly . On the other hand, we have:

Lemma 7 Let be a connected H space. Then the homotopy fiber of the multiplication is just itself, imbedded via for a homotopy inverse.

Proof: In fact, we can assume that is a CW complex, in which case is automatically an H-group, or a group object in the homotopy category. To see this, one has to show that is a group object in the homotopy category, which amounts to saying that the “shearing” map

is a homotopy equivalence. For this, we need to check that it is an isomorphism on homotopy groups by Whitehead’s theorem. It is an isomorphism on by hypothesis, and on the higher homotopy groups it corresponds to the shearing map on homotopy groups (by the Eckmann-Hilton argument). These shearing maps are clearly isomorphisms, so must be an H group.

Anyway, we can consider the sequence

which we claim is a fiber sequence. However, since the composite is nullhomotopic, we get a map from to the homotopy fiber of . If we check on homotopy groups, we can see directly that it is a homotopy equivalence.

Anyway, what all this amounts to is that the requisite diagrams all turn out to be homotopy cartesian. And then we can apply Theorem 6 above to conclude that (1) is homotopy cartesian, which finishes the proof.

3. Comments

Maybe it’s worth saying something about “Theorem 6″ above, the bit of categorical machinery used in this proof. In some sense, it is saying that homotopy colimits distribute over (homotopy) fiber products. The analogy with the word “homotopy” removed is one of the distinguishing characteristics of a Grothendieck topos. Homotopy limits and colimits should be thought of as higher categorical versions of ordinary limits and colimits, so perhaps what Theorem 6 is expressing is none other than the fact that the -category of spaces (the analog of the ordinary category of sets) is an -topos.

But I don’t actually know anything about higher topos theory, and Peter May provided on MO some reasons why I probably shouldn’t try too hard to change that in the near future. So elaborating on the previous paragraph (or acquiring the knowledge to be able to do so) will probably take me some time!

Filed under: category theory, topology Tagged: delooping, higher topos theory, homotopy colimits, infinity-categories ]]> https://amathew.wordpress.com/2012/01/11/segals-approach-to-delooping/feed/ 0 amathew 1 2 3 4 5 6 https://amathew.wordpress.com/2012/01/10/categories-and-cohomology-theories/ https://amathew.wordpress.com/2012/01/10/categories-and-cohomology-theories/#comments Tue, 10 Jan 2012 16:55:04 +0000 Akhil Mathew http://amathew.wordpress.com/?p=3085 ]]> A commutative monoid is a set together with a multiplication map and a distinguished unit element , satisfying certain identities. Let us say that we are interested in a homotopical version of this idea, especially a version of the idea of an abelian group. Then, we could try to work in the category of topological abelian groups, but this is somewhat uninteresting from the point of view of homotopy theory: every topological abelian is weakly homotopy equivalent to a product of Eilenberg-MacLane spaces. Alternatively, we could demand that one has a topological space together with a multiplication law which is commutative up to homotopy; however, as we’ve seen, this isn’t enough structure to perform a construction such as the classifying space.

Segal’s idea, in his paper “Categories and cohomology theories,” is to rephrase the definition of a commutative monoid in such a way as to require only a bunch of sets and maps with them, such that certain ones are isomorphisms. This will lead to an immediate homotopical generalization: replace “isomorphism” with “weak equivalence.”

1. Segal’s category of finite sets

We can define the category (due to Segal) of finite sets and partially defined maps as follows. The objects of are finite sets. A morphism in is the data of a subset and a map . The composition of two partially defined maps is just the ordinary composition, defined wherever it makes sense.

Suppose is a commutative monoid. We can package the data of into a functor

by sending a finite set to . Given a partially defined map between and , we get a map sending a tuple to the following tuple:

Thus, to each commutative monoid, we can associate a functor . The functors have the following properties:

1. .
2. For each , is an isomorphism, where the maps are induced by the maps defined only on .

In fact, these two properties are enough to recover and its abelian group structure. Let us be a bit more systematic. We let be the maps listed above; they are very simple, being defined only at one point (that is, ). Let us suppose we have a functor such that and such that for each , the product map

is an isomorphism. Then there is a canonically determined abelian monoid structure on , and one can phrase this as an equivalence of categories between such functors and abelian monoids.

Let’s see how we get an abelian monoid structure on if is a functor with the above two properties. In fact, the monoid structure comes from the everywhere defined map , which induces a map

where the first map comes from the postulated isomorphisms. Similarly, one gets a unit element from the map from the unique map in the category . It takes a bit of work (of chasing through the diagrams to see that one has associativity, etc.), but one can check that one in fact has a commutative monoid.

More generally, in an arbitrary category with products , we might define a strong -object to be a functor

with the above properties. Then we find that there is an equivalence of categories between -objects and abelian monoid objects in , as before.

Segal’s insight is that if we take the category of topological spaces (or simplicial sets), and replace “isomorphism” in the above with “weak homotopy equivalence,” then we get a much more flexible notion than that of a topological commutative monoid that still has a lot of structure for homotopy theory.

Definition 1 A -space is a functor satisfying the two conditions:

1. is contractible.
2. For each , the map

   

   as above, is a weak homotopy equivalence.

 2. -categories

We might also try using this definition in category theory. A commutative monoid object in category theory is a strict symmetric monoidal category. These are pretty restrictive. For instance, while any monoidal category is equivalent to a strict monoidal one, the same is not true in the symmetric case.

Definition 2 A -category is a functor satisfying the two conditions:

1. is a contractible groupoid.
2. For each , the map

   

   as above, is an equivalence of categories.

The claim is that a -category is more or less a symmetric monoidal category! Given a symmetric monoidal category , we can construct a -category as follows. The functor sends a finite set to , and given a partially defined map , we get a functor

where

Since we are working with a symmetric monoidal category, this is well-defined up to unique isomorphism. Except one should be a bit careful about compositions; might not be strictly functorial but rather 2-functorial (reflecting the fact that the tensor product is not strictly commutative or associative, but only up to natural isomorphism). I think these considerations are why a better way of phrasing the definition is to define a -category as a cofibered category over ; by the Grothendieck construction, this is approximately the same thing.

Proposition 3 A symmetric monoidal category is equivalent to the following data: A category together with a map

which makes a cofibered category over , such that the induced maps

as before are equivalences.

Here is the fiber of over . This is ultimately what leads to the definition of a symmetric monoidal -category, but I don’t really understand enough about -categories to say much more. For the purposes of this paper, the slightly more “rigid” notion of a -category (i.e. where we have a functor rather than a pseudofunctor into the category of categories) is enough.

3. From -categories to -spaces

Note in particular that the nerve of a -category is a -space; this is a consequence of the observation that an equivalence of categories induces a homotopy equivalence on the nerves (though the converse is false).

Here is a slightly different approach.

Example: Let be a category with finite coproducts. The claim is that we can obtain a -space out of , though not quite by taking the nerve (which will be contractible as has an initial object). For a finite set , let denote the poset of subsets of , and let be the category of functors

which preserve coproducts, and where the morphisms are natural isomorphisms. So, for instance, an element of is the data of three objects together with morphisms

which exhibit as a coproduct of .

The claim is that becomes now a -functor. In fact, a morphism in is precisely the same thing as a coproduct-preserving functor , so it is clear that does in fact define a functor

Now we have to check that this satisfies the additional conditions. Clearly is a contractible groupoid (it’s the category of initial objects of ). Meanwhile there is an equivalence of with in an obvious sense, since any coproduct-preserving functor is uniquely determined up to unique isomorphism by where it sends each of the singleton sets. If we apply the nerve functor to , we can get a -space.

Example: Here is an example of the previous idea. Let us take simply to be the category of finite sets (not !). We then get a -space from it. The first term is

since the first term is just the groupoid associated to , which is this.

Example: We can do the same if we replace ordinary categories by topological categories, so that their nerves are now simplicial spaces (whose geometric realizations we can take to get ordinary spaces). For instance, one can consider the category of finite-dimensional complex vector spaces with a hermitian inner product and isometric imbeddings. This clearly has coproducts, and consequently we get a -space. The value on is the nerve of the topological category of such vector spaces and isometries between them, which is equivalent to is skeleton and the nerve is thus

So this, too, is the first element in a -space.

Let be a -space. When we pass from the category of spaces to the homotopy category, we are applying a monoidal functor. This means that we get a “strong -object” in the homotopy category, in particular a commutative monoid in the homotopy category. Ultimately, we are going to want to think of the first term of -spaces as somewhat like infinite loop spaces, except that they’re not quite so because the monoid structure in the homotopy category is not necessarily a group structure. Applying a process called “group completion,” though, we will get honest loop spaces.

There is a theorem (called the “group completion theorem”) which enables one to compute what the group completion is in the cases of and ; see this article. In the first case, it is the space obtained by applying the “plus” construction to ; in the second case, it is simply . In particular, both of these should be infinite loop spaces. In both cases, there are specific reasons for it. First, is an infinite loop space by Bott periodicity, and second, is the infinite loop space of the sphere spectrum (that is, ) by a theorem of Barratt, Priddy, and Quillen.

Problem: Given a pointed space , when is of the homotopy type of a -fold loop space for some ?

One of the basic observations that one can make about a loop space is that admits a homotopy associative multiplication map

Having such an H structure imposes strong restrictions on the homotopy type of ; for instance, it implies that the cohomology ring with coefficients in a field is a graded Hopf algebra. There are strong structure theorems for Hopf algebras, though. For instance, in the finite-dimensional case and in characteristic zero, they are tensor products of exterior algebras, by a theorem of Milnor and Moore. Moreover, for a double loop space , the H space structure is homotopy commutative.

Nonetheless, it is not true that any homotopy associative H space has the homotopy type of a loop space. The problem with mere homotopy associativity is that it asserts that two maps are homotopic; one should instead require that the homotopies be part of the data, and that they satisfy coherence conditions. The machinery of operads was developed to codify these coherence conditions efficiently, and today it seems that one of the powers of higher (at least, ) category theory is the ability to do this in a much more general context.

For this post, I want to try to ignore all this operadic and higher categorical business and explain the essential idea of the delooping construction in May’s “The Geometry of Iterated Loop Spaces”; this relies on some category theory and a little homotopy theory, but the explicit operads play very little role.

1. Monads

The suspension and loop functors on the category of pointed topological spaces (at least, a convenient reformulation thereof) are adjoint. Consequently, the composition is an example of a monad. I found the definition of a monad quite confusing until I learned that a monad is essentially a monoid object in an appropriate monoidal category. Namely, the definition of a monad requires that there be a unit map

and a “composition” map

satisfying associativity and unital relations. If one considers the category of endofunctors with the monoidal structure given by composition of functors, then a monad is literally a monoid object.

An algebra over a monad is then a “module” (the term “algebra” is perhaps unfortunate) over the “monoid” ; that is, it is an object together with a “multiplication”

such that the two ways to get from (via -multiplication and , and via and ) are the same, and such that multiplication by the unit corresponds to the identity. At least, that’s how I think of it. (This deserves a post of its own.)

To be precise, we have an action

given by evaluation, and this can be interpreted as an action of the monoidal category on the category . Now if is a monad, it is a monoid in , and it makes sense for it to act on a space. This is precisely a -algebra.

2. The classifying space construction

There are a whole bunch of methods of producing a simplicial object from a monoid. Here are a few.

Given an ordinary group (or, for that matter, monoid) , there is a standard simplicial construction (due to Milnor for general topological groups) of the classifying space . Namely, one considers as a category with one object, and takes the nerve of the category. Explicitly, we have that the -simplices are given by . The degeneracy maps correspond to inserting a , while the face maps correspond to multiplying consecutive elements.

The universal cover of is given by the simplicial set , which is simplicially contractible. Here can be thought of geometrically as an infinite join of with itself. Simplicially, the -simplices are given by . It is the nerve of the groupoid whose objects are the elements and such that consists of the unique element such that . At the level of categories, we can think of this category as the “universal cover” of the groupoid from which was constructed.

So if you write everything out, not much is really happening. The simplicial set associates to a poset the set , and to a map of posets the natural map given by pulling back the coordinates. The group structure of doesn’t actually matter for constructing ; the construction actually makes sense for any set with a given basepoint, and produces a contractible simplicial set. But for a group , one notices that acts both on the left and on the right of the simplicial set (though the left and right actions are not the same).

3. The bar construction

More generally, let’s suppose is a group. Suppose acts on the right on and on the left on . Then we can form a simplicial object as follows. In degree zero, it is . In degree one, it is . In degree , it is . The face and degeneracy maps come from not only group multiplication, but from the group actions on . For instance, given , we have

This is a simplicial set, which we shall denote .

Definition 1  is the bar construction for , and .

When , this is in fact homotopy equivalent to the constant simplicial set . When , in fact we have constructed the simplicial “path space” to the simplicial set constructed above. One can write down an explicit section and a homotopy as follows. Define

In the other direction, we define

The maps described make sense in every dimension, and they are furthermore easily seen to be simplicial maps. Clearly , and it is also possible to show that is homotopic to the identity by writing down an explicit simplicial homotopy, which does not seem terribly enlightening.

4. Classifying spaces for monads

Let’s now suppose is a monad, so that it is a monoid in . Then we can say that a functor is a left module over if there is a morphism

satisfying the usual module conditions (in a monoidal category). Given an object together with a -algebra structure on (really, a “left module” structure), we can form the bar construction

This is a simplicial topological space. In dimension , we have as the space in question; the face and degeneracy maps are the same.

As before, we have:

Proposition 2 If , then is homotopy equivalent (in the category of simplicial topological spaces) to .

This is proved using the same universal formulas sketched previously in the case where we were just working with monoids and sets.

5. The delooping apparatus

The strategy is to use these bar constructions to “deloop” a space provided with sufficient construction. This is not really all that surprising, because the way of “delooping” a topological group is to form its classifying space .

Here is the main result:

Theorem 3 A topological space admitting an action of the monad is homotopy equivalent to a -fold loop space.

In fact, the key idea is that the structure of a monad action is precisely what we need to form the bar construction. We have:

Here both sides are simplicial topological spaces; is identified with the constant simplicial space. Now, all we do is pull out the to write

Next, take geometric realizations. Since as simplicial spaces,

we can take geometric realizations to get

Thus, the -fold delooping of is precisely !

Except that there are a few things to check. First, we should really check that is actually a right module over . That’s pretty easy. Whenever we have a left adjoint and a right adjoint , the composite is a monad, and is a right module over the monad. The map

is given by , where the counit of the adjunction is used. So the application of the bar construction here is actually reasonable.

What’s less obvious is that iterated loop functor (which was applied pointwise in the category of simplicial topological spaces) commutes with geometric realization. This is true under appropriate hypotheses, though the proof in May’s book somewhat technical and I don’t understand it well; it doesn’t seem to be that important to the main ideas, though. A relevant result in the simplicial context seems to be the Bousfield-Friedlander theorem as in Goerss-Jardine.

6. Other constructions

While this is a sensible delooping construction, the monad is very complicated. The use of operads has given many much simpler monads, which can more easily be shown to act on a space. In May’s book, the little cubes operad is used. I won’t go into details here, but ultimately May shows that the monad associated to this operad has the property that there is a morphism of monads

which is always a weak equivalence when applied to a connected space. As a result, when is connected and is a -algebra, then one can, as before, form the bar construction

(where is given the right -module structure from ), and this is a -fold delooping of .

A special case of May’s results is the following, when , the space is homotopy equivalent to the James construction on , i.e. the free topological monoid on , for connected.

Definition 1 The number as above such that is the Hopf invariant of .

The homotopy type of determines only on the homotopy class of , so the Hopf invariant is a homotopy invariant.

Example 1 The Hopf fibration is, by definition, the map such that the mapping cone is ; it follows that the Hopf fibration has Hopf invariant one.

The Hopf invariant is clearly identically zero for odd, but when is even the Hopf invariant is never identically zero; in fact, it defines a homomorphism

which for even has image containing the even integers. (This is where the exceptional summand in the homotopy groups of spheres comes from.)

A classical problem in topology was the following:

Question: For which does there exist a map of Hopf invariant one?

One can show that whenever (and seems to be included as a degenerate case), there is a map of Hopf invariant one. I will describe a construction next time.

Here is an example to show that there are strong restrictions on .

Example 2:  If is not a power of , there can be no map of odd Hopf invariant. In fact, in that case would be a complex whose cohomology would be where is in degree .

This, however, is impossible. In fact, if is not a power of two, the aforementioned -cohomology ring does not arise.

Proposition 2If is not a power of two and is a complex with no -cohomology between and , then the square of any element in is zero.

To prove this, we use the fact that for , we can write the cup square in terms of the Steenrod operations:

Now, by the Adem relations, the element in the Steenrod algebra is decomposable; that is, it can be written as a sum of products of smaller operations. But these smaller operations necessarily pass through for , so they are all zero.

In fact, using the Steenrod reduced powers in mod cohomology and the corresponding Adem relations, one can show that there are very strong restrictions on when a cohomology ring is a (possibly truncated) polynomial ring on one generator.

It is a theorem of Adams that in fact, are the only cases in which a map of Hopf invariant one exists. The original proof seems to have used some highly technical computations in the Adams spectral sequence. However, there is a much simpler and more transparent argument, due to Adams and Atiyah in their very enjoyable paper “K-theory and the Hopf invariant,” using a bit of K-theory.

The Adams operations

The relevant operations one needs for this argument are not the Steenrod operations in cohomology, but the Adams operations in K-theory. Let be a vector bundle. Then for each , there is supposed to be an operation such that if

then

In other words, is supposed to raise each line bundle to the th power, and is supposed to be additive. This determines . By the splitting principle, one can basically pretend that any vector bundle is a sum of line bundles, as long as one restricts oneself to operations symmetric in these individual line bundles. One can then give an explicit formula for the in terms of the exterior power operations

Anyway, in view of all this, one gets:

Proposition 3  is a natural ring homomorphism for any compact space .

For the proof, we need two more facts. The first is that for any element in the reduced K-theory , acts by multiplication by . One can see this by using the Bott periodicity theorem to identify the generator of as a power of the Hopf bundle minus one on . Then, one can reduce to showing the claim for the Hopf bundle on . For this, we have:

But the Hopf bundle satisfies the relation for ,

The higher operations can be handled similarly.

Another easy observation we’ll need is that the commute with each other. This is immediate from the description on sums of line bundles and the splitting principle.

The real reason, though, that we’re interested in these operations is that, for prime, they’re not far from raising to a power. Namely:

Proposition 4 For prime, , for .

This is exactly true if comes a line bundle, and it is preserved under sums because we are working mod . By the splitting principle, this proves the identity in general.

 The Adams-Atiyah argument

Suppose is a map, with even. The claim is that we can define the Hopf invariant solely in the setting of K-theory, which is why it is relevant at all. As usual, let’s say . Then has, as before, a CW structure with three cells, in dimensions zero, , and .

The claim is that is free abelian on two generators and . In fact, the cohomology ring is free, so by the Atiyah-Hirzebruch spectral sequence (which has torsion differentials and thus degenerates) we find that there are two generators of . We can choose the generators such that:

1. restricts to a generator of (so, in particular, it is not canonically determined).
2. is the pull-back of a generator of under the collapse map .

Under the Chern character

we find that goes to a generator of (plus possibly some extra in , and that is a generator of , even in -cohomology. This is because the Chern character is an isomorphism for even-dimensional spheres of reduced K-theory and reduced integral cohomology (this encapsulates the Bott integrality theorem earlier). Consequently, we can use naturality to argue that restricted to the -skeleton is a generator, etc.

Now, let’s apply to . We have that:

This is for the following reason: restricted to is times restricted to , by the earlier comments. For similar reasons,

Here are integers which we do not know. The main point is:

Observation: mod 2 is the Hopf invariant.

This is because is congruent to squaring mod 2. Consequently, we have in . In fact, restricted to the -skeleton is zero, so we have

If we apply the Chern character (which is integral on ), we find the analog in cohomology, that is congruent to the Hopf invariant mod 2.

So, our goal will be to show that is even. Now, we note that are very easy to compute on ; since this is obtained by pull-back from the sphere , it acts by multiplication by or .

With these preliminaries out of the way, we will simply play with the equation

and a little number theory will force to be even. Namely, from the definitions, the first side becomes

Similarly, the second side becomes

We can now equate coefficients of in both quantities. We get:

which gives

The claim is that this is going to force to be even, except in a couple of exceptional cases. Indeed, were odd, we would have that

It is a lemma now in number theory that the only possibilities for are . Granting this, the theorem is established:

Theorem 5 Suppose . Then if is a map, then all cup products in are zero.

The lemma from number theory

Actually, we don’t need the full strength of the lemma in number theory. Instead, we can modify the topological argument given above with (for odd) replacing ; everything goes through as before, except that we have

This must be true for all .

Suppose now . If we let , then

We see now that is even. In fact, must be a multiple of the order of in the group , which has order a power of two. So we can take and get

This implies that is divisible by . This implies , and the only possibilities for are . These correspond to .

to be elliptic: the associated symbol

was required to be an isomorphism outside the zero section. The goal of the index theorem is to use this symbol to compute the index of , which we saw last time was a well-defined number

invariant under continuous perturbations of  through elliptic operators (by general facts about Fredholm operators).

The main observation is that , in virtue of its symbol, determines an element of . (Henceforth, we shall identify the tangent bundle with the cotangent bundle , by choice of a Riemannian metric; the specific metric is not really important since -theory is a homotopy invariant.) In fact, we have that is the (reduced) -theory of the Thom space, so it is equivalently for the unit ball bundle and the unit sphere bundle. But we have seen that to give an element of is the same as giving a pair of vector bundles on together with an isomorphism on , modulo certain relations.

Observation: The symbol of an elliptic operator determines an element in .

1. The analytical index

There is still a problem, though. The symbol of an elliptic operator corresponds to a map , but on each fiber it is a homogeneous polynomial. By contrast, if we are interested in the -theory , we should really be interested in maps which are not given by homogeneous polynomials.

This means that one has to extend the class of operators. Rather than simply considering differential operators, one needs to consider pseudodifferential operators. These pseudodifferential operators can still be given a well-defined symbol, so it makes sense to talk about ellipticity. But the benefit is that the symbol doesn’t have to be polynomial. As a result, any element in comes from a pseudodifferential operator.

So, the main observation is:

Observation: The symbol of a pseudodifferential operator defines an element in . All elements in come from some pseudodifferential operator.

A pseudodifferential operator which is elliptic still satisfies the regularity theorem. In fact, the regularity theorem can be proved by constructing a “quasi-inverse” or parametrix (itself a pseudodifferential operator). So an elliptic pseudodifferential operator should be thought of as defining a Fredholm operator (again, in a sense which can be made precise using Sobolev spaces). In particular, we can associate to an elliptic pseudodifferential operator an “index” as before.

Here is the main result that motivates the index theorem:

Theorem 1 The index of an elliptic pseudodifferential operator depends only on the associated element in , so we get a map, called the analytical index

The idea is that the class in itself determines the homotopy class of the symbol in the space of elliptic symbols, by a general construction in K-theory. However, a homotopy class of symbols can be lifted to a homotopy of operators, and a homotopy of operators leaves invariant the index because the index is locally constant on the space of Fredholm operators.

2. The topological index

Since the analytical index defines a map

for any compact manifold , one might attempt to construct it purely topologically. In fact, this can be done, using the Thom isomorphism.

The starting point is to make into a covariant functor for imbeddings. Let be a closed imbedding of manifolds. We will define a map

At the level of elliptic operators, we are “pushing forward” an elliptic operator from to ; the crux of Atiyah-Singer’s proof is that this push-forward operation preserves the analytical index.

In order to do this, we can use the tubular neighborhood theorem to find a neighborhood of . Here is the normal bundle to in . Consequently, we have a map

which is given by the Thom isomorphism. In fact, let be the projection. Then the inclusion can be identified with the inclusion of the zero section in the complex vector bundle . As a result, we have a Thom pushforward map as claimed. Next, we use the open imbedding to push-forward ; compactly supported K-theory is covariantly functorial for open imbeddings, so this can be done. Together, we have defined the map .

Now, we have the push-forward maps, and we can use them to construct the index. If is any compact manifold, the Whitney imbedding theorem allows us to find an imbedding

for some large . As a result, we get a map

But on the other hand, the inclusion for a point defines equally a map

which is an isomorphism in view of the Bott periodicity theorem. Note that is canonically . If we take the composite, we can define:

Definition 2 The topological index is the map given by .

It requires a bit of checking that this is actually well-defined and independent of the imbedding; this follows without too much trouble from the transitivity of the Thom isomorphism, though.

3. The index theorem

The index theorem can now be stated very simply:

Theorem 3 The topological and analytical indices coincide as maps .

This gives a recipe for calculating the index of an elliptic differential (or pseudodifferential) operator on a compact manifold . Form the associated symbol homomorphism on the (co)tangent bundle . Use the “difference bundle” construction to obtain an element in . Then, compute the push-forward of this to for some imbedding , and then figure out which multiple of the (Bott) generator of it is. That number is the index of your operator.

Unfortunately this recipe (though elegant) might be a bit complicated, and one probably wants something easier to compute with: for instance, one might have given the characteristic classes of and , and one might wish to obtain the index as a number from them. In the third paper in the IEO series, Atiyah and Singer translated this into a formula in cohomology instead of K-theory. Here the point is essentially to use the Chern character map

for a space , and then to keep straight how the Thom isomorphisms in K-theory and cohomology compare to each other. This will be the subject of a future post.

 4. The proof (very sketchy)

I don’t want to present anything near the complete proof on this blog, as there a number of technical details in the analysis (which I don’t understand too well). Let me try to sketch the approach.

The topological index is a homomorphism

defined for every compact manifold. Essentially from its definition, one can quickly check that if is an imbedding, then there is a commutative diagram

Moreover, the topological index when is just the dimension map . These two properties essentiallycharacterize the topological index. If we had another transformation

which commuted with the push-forward maps above, and was the identity when , it would have to be the topological index. This is essentially formal because is then determined by its values on the spheres, and on those it is determined by its action on in view of Bott periodicity.

So most of IEO 1 is spent on establishing that the analytical index satisfies these two properties.

5. Equivariance

 The index theorem as in IEO 1 is actually stated in equivariant generality. Let be a compact Lie group acting on a compact manifold . Then is also a -space, and one can define the topological index

to the representation ring of . This is done in the same way, except now one imbeds inside an equivariant euclidean space (that is, a -representation) using the equivariant version of the Whitney imbedding theorem. Then, one needs to invoke the equivariant version of Bott periodicity, which is actually quite difficult in general. (Here one only needs it when is abelian, which is easier.)

Similarly, one can define the analytical index

which is given as follows. An element of can be represented (via the symbol) by an equivariant elliptic pseudodifferential operator . Given , one can define the equivariant index ; since are -representations by equivariance of , this makes sense.

As before, the main result is:

Theorem 4 The analytical and topological indices coincide as maps .

In IEO 1, the equivariance is built into the approach from the ground up, and in fact it is crucial to the proof. See this MathOverflow question. The reason that in the paper, equivariance is important is precisely establishing that the index commutes with the lower shriek maps above. In fact, the idea is to interpret as a suitable “product” with an equivariant K-theory class on the sphere. Since (as Atiyah-Singer show), the analytical index preserves products, one thus just has to compute some analytical indices on the sphere.

In most of the common applications, it seems that the non-equivariant version is all that one needs, though.

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 2 The 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 therational 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 3 No 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 4 There 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 5 If 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.