Topology related to Mathematical Physics, Morse Theory and Numerical Computations

与数学物理、莫尔斯理论和数值计算相关的拓扑

基本信息

批准号：
16540056
负责人：
YAMAGUCHI Kohhei
金额：
$ 2.18万
依托单位：
The University of Electro-Communications
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 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540056/
关键词：
projective variety polynomial holomorphic map homotopy type fundamental group real projective space algebraic geometry map ホモロジー群複素射影空間ポアンカレ複体多様体ホップ写像代数写像 labeled configuration space truncated configuration space m-twi

项目摘要

Previously, Professors M.Guest, A.Kozlowski and the author showed that the Atiyah-Jones-Segal type Theorem holds for spaces of holomorphic maps from the 1 dimensional complex projective space to certain family of complex projective varieties. Now he showed that a similar result holds for certain subspaces of them which are defined by using the concept of multiplicities induced from the representations of polynomials of holomorphic maps. Furthermore, he computed the fundamental groups for spaces of self-holomorphic maps on the n dimensional complex Projective spaces.Until now, we usually investigate whether AJS type Theorem holds or not for spaces of holomorphic (or algebraic) maps from one real dimensional (or complex one dimensional) spaces. In our investigation, now we can investigate whether such a problem for spaces of holomorphic or algebraic maps from high dimensional spaces. As one example, we can show that the spaces of regular maps from certain compact affine spaces into complex or real Grassmanian manifolds are homotopy equivalent of spaces of continuous maps between these spaces if these varieties Affine spaces satisfy certain conditions of vector bundles, which is one of joint works with Professor A. Kozlowski. To prove these results, we use the technique of real algebraic geometry. Moreover, we can prove that AJS type Theorem holds for such spaces by using the above Theorem. In particular, we also determine the fundamental groups of spaces of maps from m dimensional real projective space into n dimensional one when m=n-1, or m=n. Such a result can be regarded as a real version of the study investigated in the above first case.We also study the exceptional surgery from the new point view of singularity theory by using the divide theory. In particular, we study the mechanism of such surgeries and the structure of the set of exceptional surgeries.

此前，M.Guest教授，A.Kozlowski和作者表明，Atiyah-Jones-Segal类型定理可用于从1维复合物投影空间到某些复杂的投影型品种的全体形态图的空间。现在，他表明，与它们的某些子空间相似，这些结果是通过使用从全体形态图的多项式表示的概念来定义的。此外，他计算了基本组的基本组，用于n维复杂的投影空间上的自明图空间。直到现在，我们通常研究AJS型定理是否适用于一个实际尺寸（或一个复杂的一个减小）空间的全态（或代数）图的空间。在我们的调查中，现在我们可以研究来自高维空间的全态或代数地图空间的问题。作为一个例子，我们可以证明，如果这些品种贴上这些品种满足矢量捆绑包的某些条件，则常规地图从某些紧凑型仿射空间到复杂或真正的格拉马尼亚歧管的常规图空间相当于这些空间之间连续地图的均匀空间，这是与A. Kozlowski教授一起工作之一。为了证明这些结果，我们使用真实代数几何形状的技术。此外，我们可以通过使用上述定理证明AJS类型定理为此类空间保留。特别是，当m = n-1或m = n时，我们还确定了从m维真实的射击空间到n维一个尺寸的基本图。在上述第一个病例中，这种结果可以视为研究的真实版本。我们还通过使用鸿沟理论从新的点观点研究了奇异观点的特殊手术。特别是，我们研究了此类手术的机制和一组特殊手术的结构。

项目成果

期刊论文数量（74）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

The homotopy of spaces of maps between real projective spaces

实射影空间之间的映射空间的同伦

DOI：
发表时间：
2006
期刊：
J. Math. Soc. Japan 58-4
影响因子：
0
作者：
小島定吉;糸川銚;酒井隆;戸田正人;小林治;二木昭人;浦川肇;十文字正樹;山口孝男;塩谷隆;小林亮一;T.Sakai;T. Sakai;Shingo Okuyama;Kazuhisa Shimakawa;Kohhei Yamaguchi
通讯作者：
Kohhei Yamaguchi

On periodicizing functions

关于周期化函数