# Exotic spheres and chirality (Ex)


Exercise 0.1. Let $\Sigma$$\Sigma$ a homotopy $n$$n$-sphere with $n \geq 5$$n \geq 5$. Show that $\Sigma$$\Sigma$ admits an orientation reversing diffeomorphism if and only if $\Sigma$$\Sigma$ defines and element of order two in $\Theta_n$$\Theta_n$.

As a consequence, show that the boundary of the $8$$8$-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.