Research of automorphisms preserving geometric structure of manifolds

本文研究了光滑流形的微分同胚群和Lipschitz同胚群以及光滑G-流形的等变微分同胚群。我们还研究了曲面上的伪Anosov同胚。我们得到了如下结果：(1)设D(M)表示光滑流形M的微分同胚群，它们通过具有紧支撑的微分同胚同于恒等式。那么就知道D(M)是完全的。证明了当M是维界大于1的流形时，D(M)也是完美的。我们利用这个结果证明了当M是Hirzebruch-Mayer O(N)-流形时，D_G(M)是完全的。(2)设V是有限群G的表示空间，利用Sternberg的线性化定理和Tsuboi的完备性定理，计算了第一同调群H_1(D_G(M))。(3)设Γ=SL(2，Z)表示规范作用在半平面H上的模群。证明了H_1(D(H/Γ))与Γ的椭圆不动点集有关。设H*表示集合H加上Γ的尖点。证明了H_1(D(H/Γ))与Γ的椭圆不动点集有关，也与Γ的尖点集有关。(4)确定亏格g的伪Anosov同胚扩张量的最小值的问题。我们找到了重要的例子来估计每个亏格的扩张量相对于已知例子的最小值。该方法正在研究阿诺索夫流的比尔科夫截面。(5)我们召开了由科学研究助学金(http://math.shinshu-u.ac.jp/~kabe/diffeo-program.htm).部分支持的微分同态及相关领域的会议

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).