Higher Holonomies, Geometric Loop Groups and Smooth Deligne Cohomology
Higher Holonomies, Geometric Loop Groups and Smooth Deligne Cohomology
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更高的完整性、几何环群和平滑德利涅上同调
DOI：
10.1007/978-1-4612-1770-1_10
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
P. Gajer
中科院分区：
文献类型：
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作者：
P. Gajer
完整性是与平滑流形 M 上的平滑主 G 丛上的任何连接 V 相关联的结构。它为每个分段平滑环'Y: 31--t M 分配 G 的一个元素 h'V b)。如果 V 是固定的，则 h'V (CT) 仅取决于 CT 的相对同伦类，因此，它会引发同态 h'V: 7r1 (M)----+ G。众所周知， 赋值 V f---+ h'V 诱导同构 h: L (M, G, Vflat)----+ Hom (7r1 (M), G),(1) 其中 L (M, G, vflat) 是平滑主 G 丛的同构类的指向集，在 M 上有法连接。在本文中，我们研究了同构 (1) 的几种推广。首先，我们调查以下问题。如果我们将同构域 (1) 扩展到在 M 上有联系的光滑主 G 丛的同构类的点集 L (M, G, V)，那么点集 Hom (7r1 (M), G) 将发挥什么作用？可以预期，M 的循环空间 O (M) 将在这个更一般的上下文中扮演 7r1 (M) 的角色；唯一的问题是 O(M) 不是群。然而，我们可以克服这一困难，将 O (M) 除以适当的等价关系（首先由 Lefschetz 在 [Lef, Ch. V Sec. 4] 中引入）。我们用 G(M) 表示结果群，并将其称为 M 的几何环群。如果 M 是光滑流形，则 G(M) 带有可微空间结构（在 Chen 的意义上），相对于该结构，G(M) 上的群运算是可微映射。因此，G(M)是可微群。
A holonomy is a structure associated with any connection V on a smooth principal G-bundle over a smooth manifold M. It assigns to every piecewise smooth loop'Y: 31--t M an element h'V b) of G. If V is fiat, then h'V (CT) depends only on the relative homotopy class of CT, and hence, it induces a homomorphism h'V: 7r1 (M)----+ G. It is well-known that the assignment V f---+ h'V induces an isomorphism h: L (M, G, Vflat)----+ Hom (7r1 (M), G),(1) where L (M, G, vflat) is the pointed set of isomorphism classes of smooth principal G-bundles with a fiat connection over M. In this paper we investigate several generalizations of the isomorphism (1). First, we investigate the following question. What will play the role of the pointed set Hom (7r1 (M), G) if we extend the domain of the isomorphism (1) to the pointed set L (M, G, V) of isomorphism classes of smooth principal G-bundles with connection over M? One can expect that the loop space O (M) of M will play the role of 7r1 (M) in this more general context; the only problem is that O (M) is not a group. Nevertheless, one can overcome this difficulty dividing O (M) by appropriate equivalence relations (first introduced by Lefschetz in [Lef, Ch. V Sec. 4]). We denote the resulting group by G (M), and call it the geometric loop group of M. If M is a smooth manifold, G (M) carries a differentiable space structure (in Chen's sense) with respect to which the group operation on G (M) is a differentiable map. Thus, G (M) is a differentiable group.