Building-up Differential Homotopy Theory

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K18713
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 11:Algebra, geometry, and related fields
• Research Institution: Kyushu University
• Principal Investigator: IWASE Norio 九州大学, 数理学研究院, 教授 (60213287)
• Project Period (FY): 2018-06-29 – 2023-03-31
• Keywords: Diffeology / Topology / Homotopy / Loop / Algebraic / Differential / Complex

Outline of Final Research Achievements

First, we introduce a new definition of differential forms so as to obtain de Rham theorem in full generality, which can not be obtained using the genuine definition of diffeological differential forms. This also enables us to obtain the genuine de Rham theorem for smooth CW complexes.
Second, to study A∞ structure on a concatenation of paths, we first show the concatenation is smooth on a "reflexive" diffeological space, by restricting paths to satisfy the stability condition saying that paths are stable on (-∞,0] and on [1,∞).　We also introduce a "stabilised concatenation" to show that the concatenation is smooth in full generality.
Finally, we introduce a new notion of a fat smooth CW complex which enables us to conclude that a manifold is a fat smooth CW complex.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

微分空間は通常は微分不可能と考えられる対象にも微分構造を導入して解析的な操作を可能にするもので、今後の理論の展開次第では数学全体に大きなインパクトを与えうるものだと考えます。特にホモトピー論に於いてはその基礎となる対象は連続性までしか考慮されて来ませんでしたが、微分空間を考えることによりこれらは【自然に】滑らかなものとみなされます。ただ、現状ではそういった読み替えの方法が幾通りもあり、その中で真に【簡明かつ自然】なものが何なのかについて知る必要があります。本研究では【簡明かつ自然】なものとしてホモトピー論と微分構造の間に橋を架けることを希求し、その幾つかについては達成できたと考えます。