Exotic 7-sphere

Geometry Topology Student Seminar
Wednesday, April 4, 2018 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Hongyi Zhou (Hugo) – GaTech
Anubhav Mukherjee
Exotic sphere is a smooth manifold that is homeomorphic to, but not diffeomorphic to standard sphere. The simplest known example occurs in 7-dimension. I will recapitulate Milnor’s construction of exotic 7-sphere, by first constructing a candidate bundle M_{h,l}, then show that this manifold is a topological sphere with h+l=-1. There is an 8-dimensional bundle with M_{h,l} its boundary and if we glue an 8-disc to it to obtain a manifold without boundary, it should possess a natural differential structure. Failure to do so indicates that M_{h,l} cannot be mapped diffeomorphically to 7-sphere. Main tools used are Morse theory and characteristic classes.