MathDBThe smooth 4-dimensional Poincaré conjecture
The smooth 4-dimensional Poincaré conjecture
A homotopy 4-sphere is a smooth 4-manifold homotopy equivalent to the 4-sphere . Smooth 4-dimensional Poincaré conjecture. Every homotopy 4-sphere is diffeomorphic to . This is the smooth analogue of the classical Poincaré conjecture in dimension four; its status remains unresolved.
Sources & referencesView supporting material
Primary source
Rafael Torres, “A novel construction of homotopy 4-spheres and real projective 4-spaces à la Cappell-Shaneson”, arXiv:2607.15062 (2026).
Additional references
8 papers in this index state this conjecture (2009–2026). The statement above is taken from the most recent of them; the others are arXiv:2601.05425, arXiv:2412.04768, arXiv:2404.05096, arXiv:2108.10400, arXiv:1905.10938, arXiv:1707.03860, arXiv:0906.5177.
Progress summary
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