smooth structures

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?