Combinatorics of Triangulations and the Chern-Simons Invariant for Hyperbolic 3-Manifolds
Combinatorics of Triangulations and the Chern-Simons Invariant for Hyperbolic 3-Manifolds
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DOI：
10.1515/9783110857726.243
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发表时间：
1992
期刊：
影响因子：
--
通讯作者：
W. Neumann
中科院分区：
文献类型：
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作者：
W. Neumann
本文证明了关于三维伪流形三角剖分的组合学的一些结果，改进了[NZ]的结果，并应用这些结果得到了理想三角剖分的双曲三维流形的Chern-Simons不变量的一个单纯公式.将其与[MN]结合起来也给出了不变量的单纯公式。实际上，公式中的主要成分是三角剖分的理想四面体的复参数的“罗杰斯双对数”之和，但是为每个单形选择罗杰斯双对数的适当分支涉及意想不到的组合学（参见图1）。关于公式的解释，见下文注释4）。本文的组合部分（Sects. 4 - 6）是独立的和独立的利益。例如，T. Yoshida [Y2]在Culler-Shalen [CS]的精神下，使用这些组合学（[NZ]的版本）来研究字符品种和边界斜率。在本介绍的其余部分中，我们总结了ChernSimons不变量的应用。本文中的所有流形都假定是定向的。如果M是紧致的完备双曲3-流形，则它的Chern-Simons不变量CS（M）是模22定义的.如果M是非紧的，那么Bob Meyerhoff在[M]中证明了CS（M）仍然有一个模2定义良好的自然定义。设V（M）= iCS（M）+iCS（M），它是模i2 2 Z或i2 Z的良好定义.
In this paper we prove some results on combinatorics of triangulations of 3-dimensional pseudo-manifolds, improving on results of [NZ], and apply them to obtain a simplicial formula for the Chern-Simons invariant of an ideally triangulated hyperbolic 3-manifold. Combining this with [MN] gives a simplicial formula for the invariant also. In effect, the main ingredient in the formula is the sum of the “Rogers dilogarithm” of the complex parameters of the ideal tetrahedra of the triangulation, but the choice of the appropriate branch of the Rogers dilogarithm for each simplex involves unexpected combinatorics (cf. Remark 4 below for this interpretation of the formula). The combinatorial part of this paper (Sects. 4‐6) is self-contained and of independent interest. For instance, T. Yoshida [Y2] has used these combinatorics (in the version of [NZ]) to study character varieties and boundary slopes in the spirit of Culler-Shalen [CS]. In the remainder of this Introduction we summarize the application to the ChernSimons invariant. All manifolds in this paper are assumed to be oriented. If M is a complete hyperbolic 3-manifold which is compact, then its Chern-Simons invariant CS(M) is well-defined modulo 2 2 . If M is non-compact then Bob Meyerhoff has shown in [M] that there is still a natural definition of CS(M) which is well-defined modulo 2 . Let V(M) = Vol(M) + i CS(M), which is well-defined modulo i2 2 Z or i 2 Z.