Flat bundles with solvable holonomy
Flat bundles with solvable holonomy
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具有可解完整性的平丛
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发表时间：
1981
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通讯作者：
M. Hirsch
中科院分区：
文献类型：
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作者：
W. Goldman;M. Hirsch
设G是可解线性李群。我们证明了对于CW-复形M上的每个平坦的主G-丛，存在一个有限单覆盖空间：M-3M，使得p*C是平凡的主G-丛。这一结果被用来证明每个具有可解基本群的仿射流形都有一个可并行化的有限覆盖。在本文中，M表示具有基本群So的连通流形或CW-复形，G表示李群。当且仅当存在有限覆盖空间p：m-M使得p*t是平凡丛时，M上具有结构群G的丛t实际上是平凡的。如果结构群被约化为完全不连通的子群r_c_G，我们称其为平坦G-丛。如果相关主丛是平的，则向量丛是平的。下面的结果为其他人所知(例如D.Sullivan[6])，但似乎没有发表的证据。我们的目的是提供一个。定理1.设()是M上的一个平坦向量丛，它的完整群是有限生成的，且包含一个有限指标的可解子群。那么t实际上是微不足道的。证明基于定理2。设G是一个具有有限多个分支的可解李增长，它允许一个忠实的矩阵表示。设t是环M上的平坦主G-丛，其完整群是有限生成的。则(实际上是微不足道的。定理1的证明。设(是M上的平坦向量丛，由表示4：X-*GL(n；R)诱导。假设定理1的假设，我们可以转到M的有限覆盖，并假设p(4r)实际上是可解的。请注意，有限生成群中的有限指标子群是有限生成的。设G表示GL(n；R)中0(47)的Zariski闭包。由于任何代数群都有有限多个分支，且可解群的Zariski闭包是可解的，因此G满足定理2的假设，因此由4：sr-*G诱导的主G-丛实际上是平凡的。Q.E.D.我们简要地回顾了同态h：X7-+G是如何诱导平坦G丛的，我们记为h，.设p：M->；M表示泛覆盖，7r表示甲板变换群。给出XR在M×G上的对角线作用，由编辑于1980年6月4日收到，并以修订后的形式，于1980年9月24日。1980年《数学学科分类》。主55R15、57R22、22E25；辅助57R15、58H99。
Let G be a solvable linear Lie group. We show that for every flat pnncipal G-bundle { over a CW-complex M, there is a finite-sheeted covering spacep: M-3 M such that p*C is trivial as a principal G-bundle. This result is used to show that every affine manifold with solvable fundamental group has a finite covering which is parallelizable. In this note M denotes a connected manifold or CW-complex with fundamental group so and G denotes a Lie group. A bundle t over M with structure group G is virtually trivial if and only if there is a finite covering space p: M -M such that p*t is a trivial bundle. We call t a flat G-bundle if the structure group has been reduced to a totally disconnected subgroup r c G, the holonomy group of the flat bundle. A vector bundle is flat if the associated principal bundle is flat. The following result is known to others (e.g. D. Sullivan [6]) but there seems to be no published proof. Our purpose is to supply one. THEOREM 1. Let ( be a flat vector bundle over M whose holonomy group is finitely generated and contains a solvable subgroup of finite index. Then t is virtually trivial. The proof is based on THEOREM 2. Suppose G is a solvable Lie grow with finitely many components, which admits a faithful matrix representation. Let t be a flat principal G-bundle over M whose holonomy group is finitely generated. Then ( is virtually trivial. PROOF OF THEOREM 1 FROM THEOREM 2. Let ( be a flat vector bundle over M, induced by a representation 4: X -* GL(n; R). Assuming the hypotheses of Theorem 1, we may pass to a finite covering of M and assume that p(4r) is actually solvable. Note that a subgroup of finite index in a finitely generated group is finitely generated. Let G denote the Zariski closure of 0(47) in GL(n; R). Since any algebraic group has finitely many components and the Zariski closure of a solvable group is solvable, G satisfies the hypotheses of Theorem 2. Therefore the principal G-bundle induced by 4: sr -* G is virtually trivial. Q.E.D. We briefly recall how a homomorphism h: X7 -+ G induces a flat G-bundle, which we denote by h,. Let p: M -> M denote a universal covering and identify 7r as the group of deck transformations. Give Xr the diagonal action on M x G defined by Received by the editors June 4, 1980 and, in revised form, September 24, 1980. 1980 Mathematics Subject Classification. Primary 55R15, 57R22, 22E25; Secondary 57R15, 58H99.