R-Torsion and the Laplacian on Riemannian manifolds
R-Torsion and the Laplacian on Riemannian manifolds
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DOI：
10.1016/0001-8708(71)90045-4
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发表时间：
1971-10
影响因子：
1.7
通讯作者：
D. Ray;I. Singer
中科院分区：
文献类型：
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作者：
D. Ray;I. Singer
设 W 为 N 维紧致黎曼流形，K 为单纯复形，它是 W 的平滑三角剖分。K 的 Reidemeister-Franz 挠率（或 R 挠率）7 是 K 基本群的某些表示的函数。由于它是组合不变量，并且由于 W 的平滑三角剖分是等价的，因此该挠率是流形不变量。我们提出的问题是如何用解析术语来描述这个流形不变量。阿诺德·夏皮罗（Arnold Shapiro）曾经提出，可能存在一个用拉普拉斯 d 作用于 W 上微分形式的挠率公式。我们的候选 T 涉及适当拉普拉斯算子的 zeta 函数。虽然我们无法证明 T= T，但我们在本文中证明 T 是流形不变量，并提出了 T= 7 的一些证据。如果有人认为解析挠率是与 De Rham 复形相关的不变量，那么很自然会问其他椭圆复形是否也有类似的不变量。对于复流形和 &complex，确实存在这样的全纯不变量，这将是后续论文的主题。
Let W be a compact oriented Riemannian manifold of dimension N, and let K be a simplicial complex which is a smooth triangulation of W. The Reidemeister-Franz torsion (or R-torsion) 7 of K is a function of certain representations of the fundamental group of K. Since it is a combinatorial invariant, and since smooth triangulations of W are equivalent, this torsion is a manifold invariant. We raise the question as to how to describe this manifold invariant in analytic terms. Arnold Shapiro once suggested that there might be a formula for the torsion in terms of the Laplacian d acting on differential forms on W. Our candidate T involves the zeta function for appropriate Laplacians. Though we have been unable to prove that T= T, we show in this paper that T is a manifold invariant and present some evidence that T= 7.If one thinks of analytic torsion as an invariant associated to the De Rham complex, it is natural to ask whether there are analogous invariants for other elliptic complexes. For complex manifolds and the &complex, there is indeed such a holomorphic invariant, which will be the subject of a subsequent paper.