Smooth structures on collarable ends of 4-manifolds

We use Furutaʹs result, usually referred to as “108-conjecture”, to show that for any compact 3-manifold M the open manifold M × R has infinitely many different smooth structures. Another consequence of Furutaʹs result is existence of infinitely many smooth structures on open topological 4-manifolds with a topologically collarable end, provided there are only finitely many ends homeomorphic to it. We also show that for each closed spin 4-manifold there are exotic R4ʹs that cannot be smoothly embedded into it.