Relation to homotopy: Define a monoid \(G_n\) with
- Objects: smooth structures on the \(n\) sphere (identified as oriented smooth \(n{\hbox{-}}\)manifolds which are homeomorphic to \(S^n\))
- Binary operation: Connect sum
For \(n\neq 4\), this is a group. Turns out to be isomorphic to \(\Theta_n\), the group of \(h{\hbox{-}}\)cobordism classes of “homotopy \(S^n\)s”
Recently (almost) resolved question: what is \(\Theta_n\) for all \(n\)?
Application: what spheres admit unique smooth structures?