权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Research of automorphisms preserving geometric structure of manifolds

保持流形几何结构的自同构研究

基本信息

批准号：
16540058
负责人：
ABE Kojun
金额：
$ 1.15万
依托单位：
Shinshu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540058/
关键词：
diffeomorphism group Lipschitz homeomorphism group pseudo Anosov homeomorphism smooth orbifold Teichmuller space first homology equivariant diffeomorphism group smooth vector field Lipschitz同相写像可微分G-多様体 Lischitz同相写像

项目摘要

In this reseach we study the group of diffeomorphisms and Lipschitz homeomorphisms of smooth manifolds and equivariant diffeomorphisms of smooth G-manifolds. We also study pseudo Anosov homeomorphisms on the surfaces. We have the following results.(1) Let D(M) denote the group of diffeomorphisms of a smooth manifold M which are isotopic to the identity through diffeomorphisms with compact support. Then it is known that D(M) is perfect. We proved that D(M) is perfect as well when M is a manifold with boundary of dimension greater than one. We applied the result to prove that D_G(M) is perfect when M is the Hirzebruch-Mayer O(n)-manifold. Here D_G(M) denote the group of equivariant diffeomorphisms of M which are G-isotopic to the identity through diffeomorphisms with compact support.(2) Let V be a representation space of a finite group G. Using the linealization theorem by Sternberg and perfectness theorem by Tsuboi, we calculated the first homology group H_1(D_G(M)). We can apply the result to calculate H_1(D(M)) when M is a smooth orbifold.(3) LetΓ=SL(2,Z) denote the modular group which acts on the half plane H canonically. We showed H_1(D(H/Γ)) is related to elliptic fixed point set of Γ. Let H* denote the set H adding the cusp points of Γ. We proved that H_1(D(H/Γ)) is related to the elliptic fixed point set of Γ and also the cusp point set of Γ.(4) There is the problem to determine the minimal value of the dilatation of pseudo Anosov homeomorphisms of the oriented surface of genus g. We found important examples to estimate the minimal value of the dilatations for each genus g with respect to the known examples. The method is investigating the Birkov cross section of the Anosov flow.(5) We held the conference on diffeomorphism and the related fields partially supported by Grant-in-Aid for Scientific Research (http://math.shinshu-u.ac.jp/~kabe/diffeo-program.htm).

在这项研究中，我们研究了光滑流形的微分同胚和Lipschitz同胚群以及光滑G流形的等变微分同胚。我们还研究了表面上的伪阿诺索夫同胚。我们得到以下结果：(1)设D(M)表示光滑流形M的微分同胚群，它是通过具有紧支持的微分同胚的恒等式。则可知D(M)是完美的。我们证明了当 M 是边界维数大于 1 的流形时，D(M) 也是完美的。我们应用结果证明当 M 是 Hirzebruch-Mayer O(n) 流形时 D_G(M) 是完美的。这里D_G(M)表示M的等变微分同胚群，通过紧支持微分同胚，它是恒等式的G同位素。(2)设V为有限群G的表示空间。利用Sternberg的线性化定理和Tsuboi的完美性定理，我们计算了第一个同调群H_1(D_G(M))。当M是光滑环折时，我们可以应用该结果来计算H_1(D(M))。 (3)设Γ=SL(2,Z)表示规范地作用于半平面H的模群。我们证明了 H_1(D(H/Г)) 与 Г 的椭圆不动点集相关。令 H* 表示集合 H 添加 Γ 的尖点。证明了H_1(D(H/Г))与Г的椭圆不动点集和Г的尖点集有关。(4)存在亏格g定向面伪阿诺索夫同胚膨胀最小值的确定问题。我们找到了重要的例子来估计每个属 g 相对于已知例子的膨胀的最小值。该方法正在研究阿诺索夫流的比尔科夫截面。(5) 我们举办了微分同胚及相关领域的会议，部分得到科学研究资助金(http://math.shinshu-u.ac.jp/~kabe/diffeo-program.htm)的支持。