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Over the last two decades, many unexpected relations between exoticsmoothness, e.g. exotic $\mathbb{R}^{4}$, and quantum field theory were found.Some of these relations are rooted in a relation to superstring theory andquantum gravity. Therefore one would expect that exotic smoothness is directlyrelated to the quantization of general relativity. In this article we willsupport this conjecture and develop a new approach to quantum gravity called\emph{smooth quantum gravity} by using smooth 4-manifolds with an exoticsmoothness structure. In particular we discuss the appearance of a wildlyembedded 3-manifold which we identify with a quantum state. Furthermore, weanalyze this quantum state by using foliation theory and relate it to anelement in an operator algebra. Then we describe a set of geometric,non-commutative operators, the skein algebra, which can be used to determinethe geometry of a 3-manifold. This operator algebra can be understood as adeformation quantization of the classical Poisson algebra of observables givenby holonomies. The structure of this operator algebra induces an action byusing the quantized calculus of Connes. The scaling behavior of this action isanalyzed to obtain the classical theory of General Relativity (GRT) for largescales. This approach has some obvious properties: there are non-lineargravitons, a connection to lattice gauge field theory and a dimensionalreduction from 4D to 2D. Some cosmological consequences like the appearance ofan inflationary phase are also discussed. At the end we will get the simplepicture that the change from the standard $\mathbb{R}^{4}$ to the exotic$R^{4}$ is a quantization of geometry.

Smooth quantum gravity: Exotic smoothness and Quantum gravity