Smooth four-dimensional Schoenflies conjecture
Canonical statement
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For every smooth embedding \(e:S^{3}\hookrightarrow S^{4}\), each of the two closures of the components of \(S^{4}\setminus e(S^{3})\) is diffeomorphic to the standard closed \(4\)-ball \(D^{4}\).Notes
The smooth four-dimensional Schoenflies conjecture asks whether every smoothly embedded -sphere in is standard in the strongest sense: the closures of the two complementary regions should each be diffeomorphic to the closed -ball . Like the smooth Poincaré conjecture in dimension four, the modern formulation dates to around 1960, with no uniquely identifiable first source.
Outside this setting the question is settled: the smooth Schoenflies statements hold in the other dimensions, and in dimension four the locally flat topological version is a theorem. Each complementary region is a smooth homotopy -ball, and the conjecture is stronger than the smooth four-dimensional Poincaré conjecture, from which it should be kept distinct. Handle-theoretic approaches connect the problem to Generalized Property R for links [Scharlemann2010Schoenflies], and it has long featured on the standard problem lists of the field [Kirby1997ProblemsLowDimensional] [K3ProblemList2026].
A resolution requires standardizing these homotopy balls, or exhibiting a smoothly embedded -sphere in with a nonstandard complementary region. At present it is not even known that one of the two complementary smooth homotopy -balls must be standard, and the conjecture remains open.
References (3)
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