Generalized Poincaré conjecture

The conjecture that every closed manifold homotopy equivalent to a sphere is homeomorphic to that sphere. Its truth depends on dimension and on whether homeomorphism, diffeomorphism, or another equivalence is required.

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Poincaré conjecture: The three-dimensional conjecture that every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. The original three-dimensional case, proved by Grigori Perelman in 2003.
Homotopy equivalence: A map between spaces that has an inverse up to continuous deformation. The hypothesis asks a manifold to have the sphere’s homotopy type, not merely its homology.
Smooth Poincaré conjecture: The conjecture that a smooth homotopy sphere is diffeomorphic to the standard sphere. It strengthens the conclusion from homeomorphism to diffeomorphism and remains unresolved in dimension four.
Henri Poincaré: A French mathematician whose work helped establish topology as a mathematical field. His foundational work posed the original three-dimensional sphere-recognition problem.
Smale h-cobordism theorem: A theorem giving conditions under which a cobordism between manifolds is a product. It supplied the central method for proving the conjecture in dimensions at least five.
Homotopy sphere: A manifold homotopy equivalent to a sphere of the same dimension. This is the class of manifolds the conjecture seeks to identify with standard spheres.
Homology sphere: A manifold whose homology groups agree with those of a sphere. Homology equivalence does not ensure homotopy equivalence or force the standard sphere.
Stephen Smale: An American mathematician known for work in topology, dynamical systems, and geometry. His h-cobordism theorem enabled the high-dimensional proof.
Freedman’s theorem: Michael Freedman’s classification of closed, simply connected topological 4-manifolds. It establishes the topological four-dimensional case, where smooth classification remains distinct.
Cobordism: A relation between manifolds in which they form the boundary of a higher-dimensional manifold. Cobordism translates manifold classification into a framework used in high-dimensional proofs.