5-manifold

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In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.

Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected 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 Poincaré conjecture, and the 4-dimensional case is a very hard unsolved problem.

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.