Exotic Sphere - Explicit Examples of Exotic Spheres

Explicit Examples of Exotic Spheres

When I came upon such an example in the mid-50s, I was very puzzled and didn’t know what to make of it. At first, I thought I’d found a counterexample to the generalized Poincaré conjecture in dimension seven. But careful study showed that the manifold really was homeomorphic to S7. Thus, there exists a differentiable structure on S7 not diffeomorphic to the standard one.

One of the first examples of an exotic sphere found by Milnor (1956, section 3) was the following: Take two copies of B4×S3, each with boundary S3×S3, and glue them together by identifying (a,b) in the boundary with (a, a2ba−1), (where we identify each S3 with the group of unit quaternions). The resulting manifold has a natural smooth structure and is homeomorphic to S7, but is not diffeomorphic to S7. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a Morse function with just two critical points, both non-degenerate, which implies that it is topologically a sphere.

As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection of the complex manifold of points in C5 satisfying

with a small sphere around the origin for k = 1, 2, ..., 28 gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called Brieskorn spheres.