I'm currently studying some problems about the Whitehead manifold $W$ (the open 3-manifold which is contractible but not homeomorphic to $\mathbb{R}^3$). Does there exists some survey paper on its properties ?


I'm particularly interested in geometric results in the spirit of "Taming 3-manifolds using scalar curvature" by Chang, Weinberger and Yu ([MathSciNet page](http://www.ams.org/mathscinet-getitem?mr=2721617)) which shows that $W$ admits no metrics of uniformly positive scalar curvature.


Thanks.



Geometry of Whitehead manifolds.
[McMillan proved](http://www.ams.org/mathscinet-getitem?mr=131280) that any contractible 3-manifold is obtained as a union of handlebodies, each of which is homotopically trivial in the next (this generalizes the method of construction of Whitehead). 
Later he proved that there are [uncountably many topologically distinct contractible 3-manifolds](http://www.ams.org/mathscinet-getitem?mr=137105). 
There is a [survey](http://www.ams.org/mathscinet-getitem?mr=158376) of these and other results by McMillan. Even though these are old results, they are essentially state-of-the-art. The outstanding open problem was whether one of these contractible manifolds could cover a closed 3-manifold, but this is now resolved by the geometrization theorem (any closed aspherical 3-manifold is covered by $R^3$). Robert Myers obtained [partial results](http://www.ams.org/mathscinet-getitem?mr=935526) on this problem, [extended by David Wright](http://www.ams.org/mathscinet-getitem?mr=1167170), but this is superseded by geometrization. Myers also studied the [space of end-reductions](http://www.ams.org/mathscinet-getitem?mr=2199360) for such manifolds. 


Of course, the paper you are interested in needs a very minimal amount of 3-manifold topology (aside from the Poincare conjecture!), which seems to be contained in Lemma 4.1 (although I'm not sure what 3-manifold results are used in papers which they cite, in particular reference 14). 


