現代数学虎の穴 河東ゼミの教え：徹底的に調べろ４

https://en.wikipedia.org/wiki/Classification_of_manifolds
Classification of manifolds

Dimension 4: exotic
Further information: 4-manifold
Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), and surgery works topologically, but not differentiably.

Since topologically, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which (topological) 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?"

Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Similarly, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.

Dimension 5 and more: surgery
Further information: surgery theory
In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by surgery theory.

The reason for dimension 5 is that the Whitney trick works in the middle dimension in dimension 5 and more: two Whitney disks generically don't intersect in dimension 5 and above,
by general position (2+2<5).
In dimension 4, one can resolve intersections of two Whitney disks via Casson handles, which works topologically but not differentiably; see Geometric topology: Dimension for details on dimension.

https://en.wikipedia.org/wiki/Geometric_topology#Dimension
Geometric topology

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