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!
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2$\begingroup$ A reference for the fact that the product of an open contractible 3-manifold with $\mathbb R$ is homeomorphic to $\mathbb R^4$ is theorem 4 of [On contractible open 3-manifolds, E. Luft, Aequationes mathematicae (1987), Vol. 34, page 231-239], eudml.org/doc/137242. $\endgroup$– Igor BelegradekCommented Jun 8, 2022 at 1:43
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2$\begingroup$ You might find Chapter 4 of sites.math.rutgers.edu/~sferry/ps/geotop.pdf useful. $\endgroup$– skupersCommented Jun 8, 2022 at 1:55
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$\begingroup$ Thanks for providing this, I will try to figure it out. $\endgroup$– Math DiegoCommented Jun 9, 2022 at 2:10
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I would suggest a classical proof showing that the one-point compactification of $W$ is a manifold factor. See Wild wild whitehead manifold. 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).
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$\begingroup$ No wonder there exists a series of stories behind this, thanks to your explanation. $\endgroup$ Commented Jun 9, 2022 at 2:11
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$\begingroup$ @MathDiego: Just a quick comment. Andrews-Rubin's proof had great impact on the development of decomposition theory (a.k.a. Bing topology), which is different from the references I listed above. You may check the book of Bob Daverman maths.ed.ac.uk/~v1ranick/papers/daverman.pdf. Chapter 10 is dedicated to Andrews-Rubin's proof (which has more details and generality than Calegari's exposition). $\endgroup$ Commented Jun 11, 2022 at 4:32