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Smooth classifying spaces
Israel Journal of Mathematics ( IF 0.8 ) Pub Date : 2021-03-23 , DOI: 10.1007/s11856-021-2120-6
J. Daniel Christensen , Enxin Wu

We develop the theory of smooth principal bundles for a smooth group G, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define D-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling back a D-numerable bundle along smoothly homotopic maps gives isomorphic pullbacks. We then define smooth structures on Milnor’s spaces EG and BG, show that EGBG is a D-numerable principal bundle, and prove that it classifies all D-numerable principal bundles over any diffeological space. We deduce analogous classification results for D-numerable diffeological bundles and vector bundles.



中文翻译:

平滑分类空间

我们使用差分空间的框架,发展了一个光滑群G的光滑主束的理论。在给出表明为什么不能对任意主束分类的新示例之后,我们定义了D可数束,这是从拓扑结构中可数束的平滑类似物,并证明沿着平滑的同位图拉回D可数束会产生同构回调。然后,我们在Milnor空间EGBG上定义光滑结构,证明EGBGD个可数主束,并证明它对所有D分类在任何差分空间上的无数主体束。我们推导了D个数差论束和向量束的相似分类结果。

更新日期:2021-03-23
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