The Poincaré conjecture can be formulated in terms of the generalized Poincaré conjecture for different categories of manifolds. In the…

The Poincaré conjecture can be formulated in terms of the generalized Poincaré conjecture for different categories of manifolds. In the category of piecewise linear manifolds, the generalized conjecture is still an open problem for 4-dimensional manifolds. For differential manifolds, it is known to be false generally, but it is an open problem to determine in which dimensions it is true. In other words, it is not yet fully known on which spheres there exist no exotic smooth structures. The Millennium Prize was offered for the topological version of the conjecture, that remained open only for 3-dimensional manifolds during the publication of the list.

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Source: List of unsolved problems in mathematics (CC BY-SA 4.0)