So the annulus theorem has to be stated to exclude these examples, by adding some condition to ensure that S and T are well behaved.
The annulus theorem states that if any homeomorphism h of Rn to itself maps the unit ball B into its interior, then B − h(interior(B)) is homeomorphic to the annulus Sn−1×[0,1].
, where then smooth structures can be pulled back along the immersion and be lifted to covers.
The torus trick is used in Kirby's proof of the annulus theorem in dimensions
It was also employed in further investigations of topological manifolds with Laurent C. Siebenmann[1] Here is a list of some further applications of the torus trick that appeared in the literature: A homeomorphism of Rn is called stable if it is the composite of (a finite family of) homeomorphisms each of which is the identity on some non-empty open set.