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Since the first work on exotic smoothness in physics, it was folklore toassume a direct influence of exotic smoothness to quantum gravity. In thesecond paper, we calculate the "smoothness structure" part of the path integralin quantum gravity for the exotic R^4 as non-compact manifold. We discuss theinfluence of the "sum over geometries" to the "sum over smoothness structure".There are two types of exotic R^4: large (no smooth embedded 3-sphere) andsmall (smooth embedded 3-sphere). A large exotic R^4 can be produced by usingtopologically slice but smoothly non-slice knots whereas a small exotic R^4 isconstructed by a 5-dimensional h-cobordism between compact 4-manifolds. Theresults are applied to the calculation of expectation values, i.e. we discussthe two observables, volume and Wilson loop. Then the appearance of nakedsingularities is analyzed. By using Mostow rigidity, we obtain a justificationof area and volume quantization again. Finally exotic smoothness of the R^4produces in all cases (small or large) a cosmological constant.

Exotic Smoothness and Quantum Gravity II: exotic R^4, singularities and cosmology