*The primary source is the seminal 1963 differential-topology article, "Groups of homotopy spheres," by Michel Kervaire and John Milnor.*

Sean will present Section 4. The goal is to prove **Theorem 4.1:** For any fixed n > 4, there are only finitely many equivalence classes of homotopy n-spheres, where the equivalence relation is diffeomorphism up to connected sum with those **special ones** that bound a parallelizable (n+1)-manifold.

The seminar's goal is to show a *version of Theorem 1.2:* For any fixed n>4, there are only finitely many diffeomorphism classes of homotopy n-spheres. (That is, by the generalized PoincarĂ© theorem, there are only finitely many smooth structures on the standard n-sphere.) Thus Theorem 4.1 reduces us to show that there are only finitely many possibilities for the 'special ones.'

(For continuity, here's a link to the previous talk.)