Dimension 4: Exotic
For more details on this topic, see 4-manifold.4-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?"
4-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.
Read more about this topic: Classification Of Manifolds
Famous quotes containing the words dimension and/or exotic:
“God cannot be seen: he is too bright for sight; nor grasped: he is too pure for touch; nor measured: for he is beyond all sense, infinite, measureless, his dimension known to himself alone.”
—Marcus Minucius Felix (2nd or 3rd cen. A.D.)
“I did not know the woman I would be
nor that blood would bloom in me
each month like an exotic flower,
nor that children,
two monuments,
would break from between my legs....”
—Anne Sexton (1928–1974)