Why do different dimensions have different numbers of differential structures?

[saminator910]

On all manifolds dimension 1-3, there is only one differential structure per manifold, yet in higher dimensions it seems to follow no pattern. Is there a physical reason why you can construct a certain number on any given dimension? Also, what is it about dimension 4 that is so strange? Using surgery, one can construct an infinite number of differential structures on a 4 manifold, and of course most know of the Poincare conjecture. What is it that separates 4 from all the others in this way?

 

[Bacle2]

It has to see with the intersection form in dimension 4, which is a quadratic form; for every quadratic form corresponds a 4-manifold (you can actually construct it, up to homeomorphism, using Kirby calculus and other techniques), and , conversely , to (algebraically) inequivalent quadratic forms correspond non-homeomorphic manifolds. There is a correspondence between topology and algebra using Poincare duality, in that algebraic properties correspond to the actual intersection of surfaces in the 4-manifold.

Using some results from Rokhlin, you can use properties of the quadratic form to determine wether the manifold admits a differentiable structure and if this structure is unique. So, in a sense, the properties of your M^4 depend on the algebraic properties of its associated quadratic form in middle homology via Poincare Duality.

 

[lavinia]

Try reading this

http://www.ams.org/notices/201106/rtx110600804p.pdf