# Exotic spheres and chirality (Ex)

From Manifold Atlas

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.