It has been known since Milnor’s famous paper that two smooth manifolds can be homeomorphic without being diffeomorphic. Milnor showed that certain sphere bundles over were homeomorphic but not diffeomorphic to the 7-sphere . In later papers, Milnor constructed a number of additional examples of exotic spheres.

In this post, I’d like to give a detailed presentation of the argument in Milnor’s first paper.

**1. Distinguishing homeomorphic manifolds**

Suppose you have a -dimensional manifold which is known to be homeomorphic to the sphere . There are a number of criteria for this: for instance, could admit a cover by two charts, or could admit a function with only two critical points. The goal is to prove then that is *not* diffeomorphic to . Obviously the standard invariants in topology see only homotopy type and are useless at telling apart and . One needs to define an invariant that relies on the smooth structure of in some way.

It can be shown that any such is an *oriented boundary*, , for a -manifold . This is a deep fact, but in practice, the manifolds come with explicit ‘s already, and one might as well define the invariant below on boundaries. Milnor’s strategy is to define an invariant in terms of (which will depend very much on the smooth structure on ), in such a way that it will turn out to not depend on the choice of .

What can we say about ? It is a -dimensional manifold with boundary. We know that if were a*closed* manifold, then we could compute the signature

by computing the Pontryagin classes of the tangent bundle and evaluating on the fundamental class. Here is a somewhat complicated polynomial; the last term occurs to degree one as a linear term . In particular, if we write

we have

This is a fairly nontrivial divisibility relation.

However, since is not a closed manifold, the signature formula (1) is not true for , and we cannot assert (2). The existence of the boundary may introduce some fractional terms. Nonetheless, the (nontrivial) claim is that the quantity , while not necessarily integral, has residue in which is independent of the choice of manifold .

Definition 1TheMilnor invariantis defined as

this is defined for any -manifold which is a homology sphere and the boundary of some .

Here is defined as the signature of the quadratic form (cup product) on and the fundamental class of lives in relative cohomology . The Pontryagin classes of are pulled back to because in dimensions below the top dimension; this is because is a homology sphere.

It is important to use cohomology relative to the boundary since ordinary Poincaré duality does not hold for (i.e., we need to take homology and cohomology relative to the boundary).

Milnor’s observation is:

Proposition 2is actually well-defined; it does not depend on the choice of the bounding manifold .

*Proof:* To see this, suppose were two manifolds both with boundary . Then we could form the sum ; this becomes a compact, oriented manifold (without boundary) if we orient the opposite way. Then the cohomology ring of is the sum of the cohomology rings of and except in the top dimension. The fundamental class of is the “difference” of the fundamental classes of because of our choice of orientation.

Consequently, one sees that the cup square quadratic form on is the “sum” of the quadratic forms on and the opposite of the one on . Consequently, we have .

By the same reasoning, the Pontryagin classes of are determined by their restrictions to and . These are just the Pontryagin classes of and , and we have to integrate them with respect to the fundamental class of .

This means that

The last thing is a multiple of because of the signature formula applied to . This shows that when one divides by , the element of that one gets is actually well-defined.

Milnor was considering the situation where in his original paper. In this case, the -polynomial is

Consequently, we can define the invariant as

It is convenient to define from this an invariant mod 7 by multiplying by ,

This is an invariant mod 7, well defined for any homology 7-sphere.

Clearly since we can take the 8-ball as the boundary. We will compute it on certain sphere bundles and show that it does not vanish.

**2. Four-dimensional vector bundles over **

There is a Hopf fibration

Motivated by this, let’s consider the 7-manifolds that one might get by taking the sphere bundle in a four-dimensional vector bundle over .

The classification of four-dimensional vector bundles of is equivalent to the determination of

There is a double cover ; here can be identified with the group , the group of unit quaternions squared.

More explicitly, we can construct the map

by sending a pair of unit quaternions to the map , (using quaternion multiplication). Here we identify with . The kernel of this map consists of and since is central in , and consequently (by a dimension count) we have constructed the universal cover of .

It follows that we can determine :

Proposition 3. We can choose an explicit identification as follows: given , we have a map which sends a unit quaternion to the map

In fact, we have determined the structure of , and clearly of this is . The explicit identification comes from the explicit map .

**3. The sphere bundles**

Our goal now is to analyze the sphere bundles in these vector bundles over . In particular, we want to show that under specific conditions, they will be homeomorphic to the sphere.

In the previous section, we classified the four-dimensional vector bundles over , and showed that they were in bijection with . Let’s write for the vector bundle corresponding to the pair .

Proposition 4If , the sphere bundle in is homeomorphic to .

The main idea is to use the following criterion for a manifold to be homeomorphic to the sphere:

Theorem 5 (Reeb)If a compact manifold admits a function with only two critical points, it is homeomorphic to the sphere.

Any of the coordinate functions is such an example on the usual sphere.

In the present situation, we want to produce such a function on the sphere bundle in . Fortunately, is given to us fairly explicitly in coordinates. Namely, we have two charts of the sphere with transition functions, and the vector bundle is given to us explicitly with transition functions.

More specifically, we have two charts of the sphere . The transition function over is given by

Here is the coordinate on and is the coordinate on ; we have specified the coordinate transformation relating the two.

For the total space of the vector bundle , we have two charts and corresponding to the trivializations of the vector bundle on the sphere minus the north and south poles. The intersection is isomorphic to and in coordinates, we have

Here is the coordinate in the base and is the coordinate for the fiber. We have actually extended the clutching function to rather than simply on the three-sphere; this is perfectly reasonable.

The *sphere* bundle of clearly has two charts and isomorphic to , and the transition function is the same.

Now we need a function on the sphere bundle. This can be defined in coordinates, using the real part of a quaternion. The definition is:

and

We need to check that the function is actually defined everywhere and then that has only two critical points.

In fact, we can check that on the intersection, (using the fact that the real part of a quaternion is invariant under conjugation); consequently we get an honest function on the sphere bundle. It can be checked directly that there are no critical points for and has two (at ). In view of Reeb’s theorem, this completes the proof that the sphere bundles are homeomorphic to (for ).

**4. Computation of the Milnor invariant**

Now the natural thing to do is to compute on these sphere bundles, which we’ll write as . The good news is that is naturally the boundary of the ball bundle , and this is a fairly concrete manifold which we can get our hands on. There are two things to compute: the signature and the Pontryagin classes. Let’s start with the former. Throughout, we’ll assume , so that the sphere bundle is actually a topological sphere.

We have

This follows easily because deformation retracts onto . Poincaré duality for manifolds with boundary shows that the quadratic form on must be plus or minus squaring; if we choose our orientations for (and thus for !) right, we can assume

So, in order to compute , we are left with determining the Pontryagin classes of . The tangent bundle splits into the “vertical” piece (just the pull-back of ) and the “horizontal” piece (the pull-back of ). In other words, if is projection, we have

The tangent bundle does not contribute to the Pontryagin classes as it is stably trivial. So, finally, we are left with determining the Pontryagin classes of as a function of ; we may as well do it on .

Proposition 6The Pontryagin class .

*Proof:* It is easy to see that is linear in . Actually, it wasn’t easy for me when I first tried to read this paper, so let me spell it out briefly: If vector bundles and are defined by clutching functions , then is stably isomorphic to the vector bundle defined by ; this follows by a bit of matrix manipulation.

Now we need to claim that is always divisible by two, and that is in the image. This follows from the fact that is surjective onto for complex vector bundles over (a special case of a theorem of Bott, and easily checkable via Bott periodicity), and the fact that complexification

has image precisely of index 2. (Both groups are by Bott periodicity.)

To complete the proof, we need only check that . In fact, is defined by the map

Let be quaterionic conjugation.

Then

because is an anti-involution of the quaternions. But this means that and are homotopic as maps into the *infinite* orthogonal group. In fact, is not orientation-preserving, but is, and consequently conjugation by it acts trivially on .

Consequently, the vector bundles defined by and are *stably* isomorphic. This means that the Pontryagin classes are the same. This is the last step in the computation of .

Let’s now put everything together.

Theorem 7We have . If but , then is thus an exotic sphere.

*Proof:* Recall that

Here, take as the interior manifold. We saw that . The first Pontryagin class of is times the generator of , and consequently by (3), we have

Consequently,

Amazingly, the collection of manifolds homeomorphic to an -sphere (at least for ) forms a group (under connected sum), and the structure of this group can be determined in some cases: it is always finite, at least. This was a consequence of later work of Kervaire and Milnor.

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