Chern–Weil map for principal bundles over groupoids
Chern–Weil map for principal bundles over groupoids
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DOI：
10.1007/s00209-006-0004-4
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发表时间：
2004-01
影响因子：
0.8
通讯作者：
C. Laurent-Gengoux;J. Tu;P. Xu
中科院分区：
文献类型：
--
作者：
C. Laurent-Gengoux;J. Tu;P. Xu
李群胚上的主G-丛理论是一个重要的理论，它统一了流形上的主G-丛、轨道上的主G-丛以及等变主G-丛。本文研究了这类对象的微分几何，包括连通映射和完整映射。我们还为这些主丛引入了一个Chern-Weil映射，并证明了所得到的特征类与泛特征类是一致的。作为应用，我们恢复了Bott-Tu的等变Chern-Weil映射。我们还得到了Weil模型与等变上同调单纯模型之间的一个显式链映射，当流形为点时，该链映射可化为Bott-Shulman映射。
The theory of principalG-bundles over a Lie groupoid is an important one unifying various types of principalG-bundles, including those over manifolds, those over orbifolds, as well as equivariant principalG-bundles. In this paper, we study differential geometry of these objects, including connections and holonomy maps. We also introduce a Chern–Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes. As an application, we recover the equivariant Chern–Weil map of Bott–Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott–Shulman mapwhen the manifold is a point.