I am currently reading Rolfsen's "Knots and Links". At page 82 Whitehead manifold $W$ is defined and an exercise asking to show that $W\times \mathbb{R}\cong \mathbb{R}^4$ is left. Reference in Wiki says that this can be proved using Generalized Schonflies theorem, can anyone give any hint or material about this? Appreciation to your help!



How to prove the product of Whitehead manifold and $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$?
I would suggest a classical proof showing that the one-point compactification of $W$ is a manifold factor. See [Wild wild whitehead manifold](https://www.ams.org/journals/notices/201904/rnoti-p581.pdf). The proof is originally due to J. Andrews and L. Rubin, Bull. Amer. Math. Soc. 71(1965), 675-677. Once you understand that, you will see why Whitehead manifold cross R is homeomorphic to $R^4$. The key is the resulting manifold is simply connected at infinity. In general, the claim is true for any open contractible manifold (by combined forces).
For example,


J. Glimm, Two Cartesian products which are Euclidean spaces, Bull. Soc. Math. France 88 (1960), 131–135.


D. R. McMillan, Jr., Cartesian products of contractible open manifolds, Bull. Amer. Math. Soc. 67 (1961), 510–514.


J. Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481–488.


E. Luft, On contractible open topological manifolds, Invent. Math. 4 (1967), 192–
201.


E. Luft, On contractible open 3-manifolds, Aequationes Math. 34 (1987), 231–
239.


M. H. Freedman, The topology of four-dimensional manifolds, J. Differential
Geom. 17 (1982), 357–453.


However, the proofs are mainly focused on the study of the fundamental group at infinity (which is unlike the "unknotting" technique used in Andrews-Rubin's proof).


