Description

In 1956, John Milnor discovered that there exist smooth manifolds in dimension 7 which are homeomorphic (i.e. topologically equivalent) but not diffeomorphic (i.e. smoothly equivalent) to the standard 7-dimensional sphere. The existence of these exotic spheres shocked the mathematical community and led to the development of much of modern topology. Indeed, the award of several Fields Medals (including for Milnor himself) can be traced directly to Milnor's discovery.

Soon after Milnor's work appeared, geometers began asking about the extent to which the geometry of exotic spheres resembles that of the standard sphere, which is the basic example of a manifold with positive (sectional) curvature. In 1974, Detleff Gromoll and Wolfgang Meyer found a novel construction of one particular exotic sphere in dimension 7 and, as a byproduct, were able to conclude that this Gromoll-Meyer exotic 7-sphere is non-negatively curved. It is now known that the Gromoll-Meyer exotic 7-sphere is positively curved almost everywhere, but it remains open whether it is positively curved.

This project will study the Gromoll-Meyer exotic 7-sphere. We will see how it relates to Milnor's original discovery, as well as how its construction yields information about its geometry. Along the way, we will study the basic Riemannian geometry of Lie groups and their quotients. If we get through all that, we can look at what is known about constructions and the geometry of other exotic spheres.

Prerequisites

• Topology III

• Algebra II
• Differential Geometry III

Corequisites

• Riemannian Geometry IV
• Algebraic Topology IV

Some Resources

• Detlef Gromoll and Wolfgang Meyer, An Exotic Sphere With Nonnegative Sectional Curvature, Annals of Mathematics, Vol. 100, No. 2 (1974), 401-406

• John Milnor, On Manifolds Homeomorphic to the 7-Sphere, Annals of Mathematics, Vol. 64, No. 2 (1956), 399-405

• Manfredo P. do Carmo, Riemannian Geometry, Birkhäuser 1992