From what I understand, an exotic n-sphere is a manifold which is homeomorphic to the n-sphere but not diffeomorphic to it. Now I have read that there are no exotic 2-spheres. But isn't something like a tetrahedron an example of a manifold which is homeomorphic to the sphere, but not diffeomorphic ? (Because of the corners and edges.) What am I missing ?
A exotic sphere is (by definition) a differentiable manifold. So if you want to consider the tetrahedron, you have to specify, what differentiability at one of the edges means. As soon as you specified this, you will just get the 2-sphere.