Model and Set-Theoretic Aspects of Exotic Smoothness Structures on $\mathbb{R}^4$
Jerzy Król
备注：
32 pages, 2 figures, to appear in: At the Frontiers of Spacetime: Scalar-Tensor Theory, Bell's Inequality, Mach's Principle, Exotic Smoothness, ed. T. Asselmeyer-Maluga (Springer, 2016), in honor of Carl Brans's 80th birthday
Model-theoretic aspects of exotic smoothness were studied long ago uncovering unexpected relations to noncommutative spaces and quantum theory. Some of these relations were worked out in detail in later work. An important point in the argumentation was the forcing construction of Cohen but without a direct application to exotic smoothness. In this article we assign the set-theoretic forcing on trees to Casson handles and characterize small exotic smooth $R^4$ from this point of view. Moreover, we show how models in some Grothendieck toposes can help describing such differential structures in dimension 4. These results can be used to obtain the deformation of the algebra of usual complex functions to the noncommutative algebra of operators on a Hilbert space. We also discuss the results in the context of the Epstein-Glaser renormalization in QFT.
On the origin of inflation by using exotic smoothness
在这篇论文中，我们讨论了一个拓扑结构为S^3 x R的时空，但它具有不同的光滑结构，导致了一个暴胀的几何模型，称为几何暴胀。特别是这个时空是不是全球双曲，我们得到一个时间线的空间拓扑变化从3球的同源3球和回来。时空的拓扑保持不变。在这个时空的无限可能的光滑结构中，我们选择了一个由纽结8_{10}构造的具有双曲几何的同调3-球，即允许负标量曲率的齐次度量。我们讨论了FLRW宇宙学由于拓扑结构的改变而引起的加速膨胀。与其他膨胀模型不同，这个过程在有限的时间后停止。或者，拓扑变化也可以由SU（2）值标量场描述。然后我们计算膨胀率（超过60个e-折叠）和能量时间尺度。与物质的耦合也被几何地解释，并且自然地获得再加热过程（以及在膨胀期间的过冷膨胀）。该模型只依赖于一个参数，即同调3球的拓扑不变量，并假设一个普朗克尺度的S^3拓扑宇宙。讨论了模型对初始态的依赖性和量子涨落的几何解释。
