Reidemeister torsion and the laplacian on lens spaces
Reidemeister torsion and the laplacian on lens spaces
复制标题
雷德迈斯特挠率和透镜空间上的拉普拉斯
DOI：
10.1016/0001-8708(70)90018-6
复制
发表时间：
1970
期刊：
影响因子：
--
通讯作者：
D. Ray
中科院分区：
文献类型：
--
作者：
D. Ray
设g是奇数维33的球面的旋转，使得对于某个整数p 3 3 3，gp是单位元，而g”对于0< K<p没有不动点。在g的作用下，对S1-r的点的识别产生透镜空间L1-r（g）。由于L_（1 - 1）（g）的Betti数不依赖于旋转g，因此透镜空间是研究流形拓扑分类的好例子。透镜空间的分类是在1935年通过引入Reidemeister扭转完成的。粗略地说，复形K的挠率是通过将K作为基本整环嵌入到泛覆盖复形R中而得到的。K中K的单元e的边界则由K的单元通过甲板变换的平移ge'的组合给出。考虑K的胞元作为优选基，这产生了边界矩阵，其元素位于基本群G的群环中。群G到模1的复数的每个同态将边界矩阵变换为复矩阵。然后寻找这些边界矩阵的不变量下细分和下变换的拓扑允许的变化的基础。Franz（[2]）发现，除了Betti数外，只有一个不变量，即挠率。他用扭转证明，两个透镜空间是组合等价实际上是等距的，使旋转定义他们是共轭的。挠率由一个公式定义，该公式涉及从边界矩阵构造的某些行列式。鉴于德拉姆定理有关的组合上同调的一个封闭的定向流形的上同调的微分形式，问题出现了是否有一个类似的公式涉及外微分的形式。
Let g be a rotation of a sphere &-r of odd dimension 33, such that, for some integer p 3 3, gp is the identity, while g” has no fix points for 0< K< p. Identification of the points of S,,-, under the action of g produces the lens space L,,-,(g). Since the betti numbers of L,,-,(g) do not depend on the rotation g, the lens spaces make good examples in studying the topologic classification of manifolds. The classification of lens spaces was accomplished in 1935 ([2]) by the introduction of the Reidemeister torsion. Roughly speaking, the torsion of a complex K is obtained by embedding K as a fundamental domain in the universal covering complex R. The boundary in K of a cell e of K is then given by a combination of translates ge’of cells of K by deck transformations. Thinking of the cells of K as a preferred base, this gives rise to a boundary matrix whose entries lie in the group ring of the fundamental group G.Each homomorphism of the group G into the complex numbers of modulus one transforms the boundary matrix into a complex matrix. One then looks for invariants of these boundary matrices under subdivision and under transformation by topologically permissible change of base. Franz ([2]) f ound that there was just one invariant, the torsion, besides the betti numbers. He used the torsion to prove that two lens spaces which are combinatorially equivalent are actually isometric, so that the rotations defining them are conjugate. The torsion is defined by a formula involving certain determinants constructed from the boundary matrices. In view of de Rham’s theorem relating the combinatorial cohomology of a closed oriented manifold to the cohomology of differential forms, the question arises of whether there is an analogous formula involving the exterior differential for forms.