扭丛与扭K理论
Twisted bundles and twisted K-theory
摘要：
我们在这里提供了一种更直接的扭曲K理论方法，该方法基于扭曲向量丛（有限维或无限维）和扭曲主丛的概念。这与代数丛上的模的torsors和丛的经典概念密切相关。扭曲K理论被简单地定义为扭曲向量丛的Grothendieck群（具有给定的扭曲）。通常对向量丛的运算（外幂、Adams运算）很容易推广到这个扭曲的框架。分次扭曲的K理论，这是重要的定义和证明的Thom同构在这个框架中，定义了相同的形式主义。我们还定义了一个扭曲的陈特征标，目标是“扭曲的上同调”。虽然在文献中有许多定义这个字符，我们的方法是更基本的，是基于经典的定义的陈-韦伊理论，通过连接和曲率。这样，我们得到的显式公式非常符合Steenrod坐标丛的精神。
英文摘要：
We offer here a more direct approach to twisted K-theory, based on the notion of twisted vector bundles (of finite or infinite dimension) and of twisted principal bundles. This is closeely related to the classical notion ot torsors and bundles of modules over an algebra bundle. Twisted K-theory is simply defined as the Grothendieck group of twisted vector bundles (with a given twist). The usual operations on vector bundles (exterior powers, Adams operations) are easily extended to this twisted framework. Graded twisted K-theory, which is important for the definition and proof of the Thom isomorphism in this framework, is defined with the same formalism. We also define a twisted Chern character with target "twisted cohomology". Although there are many definitions of this character in the literature, our method is more elementary and is based on the classical definitions of the Chern-Weil theory, via connections and curvatures. In this way, we get explicit formulas very much in the spirit of Steenrod's coordinate bundles.
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