There are many theorems in the differential geometry literature of the following sort. Let M be a complete Riemannian manifold with some conditions on various curvatures, diameters, volumes, etc. Then M is homotopy equivalent to a finite CW complex, or M is the interior of a compact, topological manifold with boundary. At first glance it seems unlikely that such theorems have anything to say about smooth manifolds homeomorphic to R-4. However, there is a common theme to all the proofs which forbids the existence of such metrics on most (and possibly all) exotic R4's.
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