Moduli theory of non linear elliptic operators over non compact manifolds

2016 Fiscal Year Final Research Report

• Project/Area Number: 25287009
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Kato Tsuyoshi 京都大学, 理学(系)研究科(研究院), 教授 (20273427)
• Co-Investigator(Kenkyū-buntansha): 木田 良才 東京大学, 数理(科)学研究科(研究院), 准教授 (90451517)
• Co-Investigator(Renkei-kenkyūsha): Kida Yoshikata 東京大学, 数理科学研究科, 准教授 (90451517) ; Oguni Shin-ichi 愛媛大学, 理工学研究科, 准教授 (00549446) ; Fukaya Tomohiro 首都大学, 東京理工学研究科, 准教授 (40583456) ; Tsukamoto Masaki 京都大学, 理学研究科, 准教授 (70527879) ; Matsuo Shinichiroh 名古屋大学, 多元数理科学研究科, 准教授 (40599487)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: ゲージ理論 / 非可換幾何学 / モノポール写像 / 普遍被覆空間

Outline of Final Research Achievements

I have constructed a monopole map over the universal covering space of a compact oriented smooth four manifold. We apply the infinite dimensional Bott periodicity by Higson-Kasparov-Trout. In particular its degree was given when the linearized map is isomorphic, as an element in the equivariant E theory. It produces a homomorphism between K group of C* algebras related to the group ring. It corresponds to a covering version of the Bauer-Furuta degree.
As an application, we proposed an aspherical 10/8 inequality for spin classifying 4 manifolds. We have also verified that it certainly holds for large class of 4 manifolds which includes complex minimal surfaces of general type.

Free Research Field

幾何学