Torsten,
At the risk of sounding like a crank, I have done some thinking about this matter of exotic manifolds, which I thought I might relay. If this is wrong that is fine; if this is wrong I'd rather know this is flawed than to think otherwise. My essay discusses noncommutative geometry, which implies manifold structure exhibits quantum fluctuations on a very small scale. As one approaches the Planck length manifold structure becomes indefinable or difficult to define. Fluctuations of coordinates Δq_iΔq_j ~ Għ/c^3, due to nonzero commutators [q_i, q_j] = Għ/c^3, are such that any coordinate q'_i = q_i Δq_i may be arbitrarily large, leading to a breakdown (or uncertainty) in differentiability.
This is in contrast to your approach which involves smooth manifolds that are exotic, or which are homeomorphic to R^4, but not diffeomorphic. These seem to be completely opposite conclusions about the nature of quantum space or spacetime. However, in both cases we have a subtle issue with differentiability. It also occurred to me in reading your essay that you consider a manifold with the D^2xT^2, similar to a Casson handle, removed and replaced with S^3\(D^2xK)xS, for K = knot. S^3\(D^2xK)xS is the space left behind when the knot is lifted out. The manifold M\(D^2xT)xS U S^3\(D^2xK)xS is then exotic. This is then argued for a case with R^4 such that it is foliated by S^3xR by a parameter t. A similar structure exists where the volume and curvature are a Chern-Simon's invariant.
The CS action is given by a quantum group or knot structure. The knot structure is a given by S_{cs} = ∫_γ L, for L = A/\dA (2/3)A/\A/\A. The action for this group is determined by the group SL(2,C) which defines linear fractional transformations
f(z) = (az b)/(cz d)
for a = b= d and c = 1 this is the inversion. T-duality in string theory is based around this, which is a duality between the frequencies of a closed string on a space, and the tension of the string wrapped around that space. There is a theorem by Connes which states there is a map or functor between K-theory of a quantum group or the set of all crossings like the Jones polynomial determined by the CS Lagrangian and cohomology of a C* algebra. The question is then whether these two are related.
The inversion between large and small, and the physical expectation is that a manifold will have smooth structure on a large scale and fluctuating structure on a small scale. This might suggest there is a dual description of a quantum manifold. One is of a smooth structure with a "quantization" according to units of volume and curvature, and the other according to noncommutative structure. The two are related by scale factors or a T-duality type of structure.
The four manifold is Poincare dual to a 7 dimensional space in 11 dimensional superspace, or 6 dimensional space in 10 dimensions. The compactification of the six dimensions is a Calabi-Yau space CY = T^6, K3xK3 or some similar Ricci flat space, which in 7 dimensions is CYxS. Closed strings on the four dimensional spacetime are wrapped around the CY manifold. The T-duality between string modes and string tension I think is a Poincare duality with the above T-like-duality that I conjecture.
Cheers LC
