Classification of contractible 4-manifolds Is there a general homeomorphism classification of contractible topological 4-manifolds (possibly with boundary or noncompact)? 
In the compact case, any such manifold has a homology 3-sphere as its boundary (according to Wikipedia). Homology 3-spheres can be classified (via geometrization of 3-manifolds), so maybe there is hope for a classification of the interiors also.
 A: If you work in topological category and assume that your manifolds are compact then any two compact contractible 4-manifolds $W_1, W_2$ with homeomorphic boundaries $\partial W_1, \partial W_2$ are homeomorphic. This follows for instance from the main classification result of Richard Stong: 
R. Stong, Simply-connected 4-manifolds with a given boundary. 
Topology Appl. 52 (1993), no. 2, 161–167. 
Roughly speaking, Stong extended Freedman's classification theorem to the case of compact simply-connected 4-manifolds with boundary. 
(The special case of contractible manifolds might have been known earlier, I am not sure.) 
Thus, if you think that 3-dimensional integer homology spheres are "classified" then so are compact topological contractible 4-manifolds. 
(Personally I regard the classification problem of integer homology spheres which are hyperbolic 3-manifolds to be hopelessly complicated. But modulo this problem, yes, we "know" what 3-dimensional homology spheres are.) Of course, a classification up to diffeomorphism of smooth 4-manifolds with the given boundary is well beyond reach at this point. 
