Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first classified by Dennis Barden and another proof was later given by A. V. Zhubr. Rather surprisingly, this turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by Freedman (1982) in the topological case, but is a very hard unsolved problem in the smooth case.
Indeed, in dimension 5 smooth classification is governed by classical algebraic topology, namely, two simply connected 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving linking form and the second Stiefel–Whitney class. Moreover any such isomorphism is induced by some diffeomorphism.
- 1-connected 5-manifolds at the Manifold Atlas.
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