Smooth Poincare Conjecture: Is There an Exotic Four-sphere?

Smooth Poincare Conjecture: Is There an Exotic Four-sphere?

-
Zhong Tao Wu
Fine Hall 314

After briefly reviewing the history of (smooth) Poincare conjecture, we shall focus on the case of four-dimensional sphere. We would like to discuss a construction due to Gluck that provides many candidates of exotic four-spheres, i.e., manifolds homeomorphic but not diffeomorphic to the standard four-sphere. We will prove that Gluck's manifolds are indeed homeomorphic to four-sphere and then sketch an argument by Freedman et al. that could potentially lead to the discovery of exotic four-spheres.