Since I already gave a talk about Milnor’s paper on constructing exotic -spheres (which are smooth manifolds that are homeomorphic but not diffeomorphic to the standard -sphere) anyways, I figured it was a reasonable topic for a blog entry. The result is also pretty cool, as it very much disagrees with our intuition of how spheres should behave, based on examples in lower dimensions, and furthermore is given by a construction that is reasonably tangible, (the ones that Milnor constructs are -sphere bundles over the -sphere for which an explicit classifying map is given, as well as an explicit Morse function to show that it is in fact a topological -sphere) rather than something arising from invoking the axiom of choice. Of course, there are nonintuitive results about higher dimensions that can be more simply stated- for example, if we embed unit spheres in the corners of an -dimensional cube with all side lengths equal to , we can ask how large the sphere in the center is. Since it must be tangent to all of the other spheres, we can simply use the diagonal of the cube to calculate its length. By the pythagorean theorem, the length of the diagonal of the cube is , so subtracting the diameters of the two spheres, we find that the inner sphere has diameter equal to . However, this means that when , the inner sphere has diameter , and therefore is tangent to the hypercube at all of its faces. Even weirder still is the situation when . In this case, , so the diameter of the sphere is larger than the minimal distance across the cube, telling us that near enough to the center of the faces, the sphere is actually protruding out of the cube.
Now, for to be an exotic -sphere, must satisfy two properties: (1) is not diffeomorphic to , and (2) is homeomorphic to . This at first seems very counterintuitive- how can we possibly find a homeomorphism between smooth manifolds that is both nice enough to write down, but also not a diffeomorphism?
We know that any diffeomorphism of smooth manifolds naturally induces an isomorphism of tangent bundles , as gives us an isomorphism of tangent spaces at each point. Thus, we want but .
It is well known that for an -dimensional manifold, if there is a smooth map such that has exactly two critical points, (both of which are nondegenerate) then is homeomorphic to an -sphere. This is the criterion that we will use to check that our constructed manifold is homeomorphic to .
We now consider manifolds that are given as -sphere bundles over . These are classified by maps , in other words, by elements of . Note that this is isomorphic to , by the following correspondence: for each , we have the map , where quaternion multiplication is understood on the right. Let be the -sphere bundle corresponding to a given pair . If we let be the standard generator for , it turns out that the Pontryagin class .
For be such that , define to be the total space of the bundle . Then, it turns out that satisfies our previous condition for being homeomorphic to . Furthermore, given the Pontryagin class, we find that , and therefore is not diffeomorphic to the standard -sphere, as desired.