Understanding exotic spheres from the viewpoint of global singularity theory of smooth maps

从光滑映射全局奇点理论的角度理解奇异球体

• 批准号: 18F18752
• 负责人: 佐伯 修
• 金额: $ 1.41万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2018
• 资助国家: 日本
• 起止时间: 2018-11-09 至 2021-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18F18752/
• 关键词: fold singularities; rational homologyspheres; linking form; intersection spaces; toric varieties; bordism of smooth maps; B_2 singularity; fold singularity; smooth map germ; homotopy sphere; SKK-group; signature; intersection space; cobordism

In a recent preprint (arXiv: http://arxiv.org/abs/2009.05928), we studied the existence and construction problems for special generic maps of rational homology spheres. The novelty of our approach is to consider the torsion subgroup of the integral homology of such manifolds. We showed that if a rational homology sphere of odd dimension n = 2k + 1 > 4 admits a special generic map into a Euclidean space of dimension < n, then the cardinality of its integral homology group of degree k is a square. On the one hand, we showed that any square can can be realized in our homological condition. On the other hand, there are examples of rational homology spheres that do not satisfy our homological condition. Our results paved the way to a subsequent project, in which we study special generic maps of highly connected manifolds in terms of the linking form, which is a torsion analog of the intersection form. In another project, we developed a new approach to intersection spaces that is based on Sullivan's PL polynomial differential forms. Our result implies uniqueness of the rational cohomology ring of intersection spaces. This result is a new discovery in the research field, and we published the case of isolated singularities. In ongoing work, we generalize our approach along the construction of Agustin and Fernandez de Bobadilla to a class of singular spaces of arbitrary stratification depth including toric varieties. Moreover, in joint work with T. Essig, we are finalizing a preprint about the construction of a fundamental class for intersection spaces in stratification depth two.

在最近的一篇预印本（arXiv: http://arxiv.org/abs/2009.05928）中，我们研究了有理同调球的特殊一般映射的存在性和构造问题。该方法的新颖之处在于考虑了这类流形的积分同调的扭转子群。证明了如果奇数维数n = 2k + 1 > 4的有理同调球允许一个特殊的泛型映射到维数< n的欧几里得空间中，则其k次整同调群的基数是平方。一方面，我们证明了在我们的同调条件下，任何平方都可以被实现。另一方面，也有不满足我们的同调条件的有理同调球的例子。我们的结果为随后的项目铺平了道路，在该项目中，我们研究了高连接流形在连接形式方面的特殊泛型映射，这是交叉形式的扭转模拟。在另一个项目中，我们开发了一种基于Sullivan的PL多项式微分形式的相交空间的新方法。给出了交空间的有理上同环的唯一性。这个结果是研究领域的一个新发现，我们发表了孤立奇点的情况。在正在进行的工作中，我们沿着Agustin和Fernandez de Bobadilla的构造将我们的方法推广到一类具有任意分层深度的奇异空间，包括环面变异。此外，在与T. Essig的联合工作中，我们正在完成一份关于在分层深度2中的相交空间的基本类的构建的预印本。

项目成果

Linking between singular locus and regular fibers

奇异位点和规则纤维之间的连接

• DOI: 10.5427/jsing.2020.21n
• 发表时间: 2020
• 期刊: Journal of Singularities
• 影响因子: 0.4
• 作者: 梶原健司，D.Triadis;P. Broadbridge;丸野健一;Kenji Kajiwara;梶原健司;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;梶原健司;梶原健司;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;梶原健司;Kenji Kajiwara;Osamu Saeki
• 通讯作者: Osamu Saeki

A signature invariant for stable maps of 3-manifolds into surfaces

3-流形到表面的稳定映射的签名不变量

• 发表时间: 2019
• 期刊: Rev. Roumaine Math. Pures Appl.
• 作者: 梶原健司，D.Triadis;P. Broadbridge;丸野健一;Kenji Kajiwara;梶原健司;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;梶原健司;梶原健司;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;Kenji Kajiwara;梶原健司;Kenji Kajiwara;Osamu Saeki;大槻知忠,高田敏恵;Saeki Osamu;Saeki Osamu
• 通讯作者: Saeki Osamu

ELIMINATION OF DEFINITE FOLD. II

消除明确的折叠。

• DOI: 10.2206/kyushujm.73.239
• 发表时间: 2019
• 期刊: Kyushu Journal of Mathematics
• 影响因子: 0.4
• 作者: Gupta;A. and Matsumoto;J.;金子 拓;Osamu Saeki
• 通讯作者: Osamu Saeki

Cobordism theory of Morse functions and applications

莫尔斯函数的协边理论及其应用

• 发表时间: 2020
• 作者: Mueller L. Felipe;Wrazidlo Dominik J.;Dominik Wrazidlo;Osamu Saeki;Osamu Saeki;Osamu Saeki;O. Saeki;O. Saeki;D.J. Wrazidlo;D.J. Wrazidlo
• 通讯作者: D.J. Wrazidlo

The Chromatic Brauer Category and Its Linear Representations

半音布劳尔范畴及其线性表示

• DOI: 10.1007/s10485-020-09619-5
• 发表时间: 2020
• 期刊: Applied Categorical Structures
• 影响因子: 0.6
• 作者: Mueller L. Felipe;Wrazidlo Dominik J.
• 通讯作者: Wrazidlo Dominik J.