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.
[1] Simply connected compact 5-manifolds were first classified by Stephen Smale[2] and then in full generality by Dennis Barden,[3] while another proof was later given by Aleksey V.
In dimension 5, the smooth classification of simply connected manifolds is governed by classical algebraic topology.
Namely, two simply connected, smooth 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving the linking form and the second Stiefel–Whitney class.