What's so messed up about d dimensions?

This is going to be a big list so I will build it up gradually.

Dimension 2

• Orientable, metrizable surfaces can be given a Riemann Surface structure which are conformally equivalent to sphere, plane or hyperbolic plane or quotients thereof.
• is one of the two possible spheres which admit an almost complex structure
• Highest dimension in which continuous symmeteries cannot be spontaneously broken.

Dimension 3

• Geometry of finite volume hyperbolic manifolds of at least this dimension is determined by the fundamental group. The finite isometry group left is given by outer automorphisms of the fundamental group.

Dimension 4

• Having a h-cobordism between two simply connected smooth 4-manifolds does not give you an isomorphism between them.
• This is the only dimension in which there are exotic versions of and there are infinitely many to boot.
• The double cover of the conformal group of 3+1 dimensions is Spin(4,2)=SU(2,2) allowing twistors
• Can solve Ising model in 3+1 with the usual bosonic variables
• No Hair theorem in 3+1 dimension
• You begin to be able to get away with mean field theory in Ising

Dimension 6

• is one of the two possible spheres which admit an almost complex structure
• Last Superconformal Algebra

Dimension 8

• There is a unique even self-dual lattice. It is the one for the   root system.

Dimension 10

• Conformal anomaly in superstrings cancels

Dimension 11

• Appears at strong coupling