Exotic spheres and chirality (Ex)
Recall that denotes the group of h-cobordism classes of manifolds which are homotopy equivalent to the -sphere. For this is the same as the group of oriented diffeomorphism classes of such manifolds.
Exercise 0.1. Let a homotopy -sphere with . Show that admits an orientation reversing diffeomorphism if and only if defines and element of order two in .
As a consequence, show that the boundary of the -dimensional E_8-manifold does not admit an orientation reversing diffeomorphism.
Remark 0.2. Note that manifolds admitting no orientation reversing diffeomorphism are often called chiral.