Classification of Manifolds - Overview By Dimension

Overview By Dimension

• Dimensions 0 and 1 are trivial.
• Low dimension manifolds (dimensions 2 and 3) admit geometry.
• Middle dimension manifolds (dimension 4 differentiably) exhibit exotic phenomena.
• High dimension manifolds (dimension 5 and more differentiably, dimension 4 and more topologically) are classified by surgery theory.

Thus dimension 4 differentiable manifolds are the most complicated: they are neither geometrizable (as in lower dimension), nor are they classified by surgery (as in higher dimension or topologically), and they exhibit unusual phenomena, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Notably, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.

One can take a low dimensional point of view on high dimensional manifolds and ask "Which high dimensional manifolds are geometrizable?", for various notions of geometrizable (cut into geometrizable pieces as in 3 dimensions, into symplectic manifolds, and so forth). In dimension 4 and above not all manifolds are geometrizable, but they are an interesting class.

Conversely, one can take a high dimensional point of view on low dimensional manifolds and ask "What does surgery predict for low dimensional manifolds?", meaning "If surgery worked in low dimensions, what would low dimensional manifolds look like?". One can then compare the actual theory of low dimensional manifolds to the low dimensional analog of high dimensional manifolds, and see if low dimensional manifolds behave "as you would expect": in what ways do they behave like high dimensional manifolds (but for different reasons, or via different proofs) and in what ways are they unusual?