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Study on Fiberwise A-infinity Structures

Research Project

Project/Area Number 19K14535
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11020:Geometry-related
Research InstitutionKyushu University

Principal Investigator

Tsutaya Mitsunobu  九州大学, 数理学研究院, 准教授 (80711994)

Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Keywordsホモトピー論 / ファイバーワイズホモトピー論 / A無限大空間 / crossed module / A無限大構造 / ホモトピー正規性 / ホモトピー可換性 / 高次ホモトピー構造 / An写像 / 高次ホモトピー正規性 / ファイバーワイズホモトピー / uniform Roe algebra / ファイバーワイズA無限大構造 / ユニタリ群 / 代数的位相幾何学 / 位相的複雑さ
Outline of Research at the Start

本研究の目的はファイバーワイズA無限大空間に関する基礎理論の構築と,関連する不変量の計算手法の確立である.
高次ホモトピー構造は写像の族で記述されるため,一般に障害などの不変量が記述しづらいが,通常のA無限大空間の場合には,射影空間とよばれる空間が関手的に構成され,これを用いて古典的なホモトピー論の手法で不変量が記述される.本研究では射影空間のファイバーワイズ版についての基礎理論の構築を目的とする.
さらに,射影空間のコホモロジーの計算を通して,位相的複雑さなどのファイバーワイズホモトピー論の不変量との関係の解明および計算手法を確立することも目的とする.

Outline of Final Research Achievements

The most important result of this project is the development of the theory of higher homotopy normality using fiberwise A-infinity structures. Though higher homotopy normality has been studied by several people, there are no theory of ``essentially higher'' homotopy normality. The theory established in this project is a candidate for such theory. It enables us to determine when a given homomorphism has higher homotopy normality by the classical technique in the fiberwise homotopy theory. Indeed, the p-local higher homotopy normality of the inclusions SU(m) -> SU(n) are determined for some m,n and p.
We also obtained some results on the homotopy type of the unitary groups of some uniform Roe algebras in a joint work. Comparing to Roe algebras, uniform Roe algebras tend to have huge K-theory. We could determine the homotopy type of the unitary groups of uniform Roe algebras on Z and Z^2.

Academic Significance and Societal Importance of the Research Achievements

高次ホモトピー正規性はこれまでにも研究されてきたが,「本質的に高次の」ホモトピー正規性の理論は得られていなかった.得られた理論では古典的なファイバーワイズホモトピー論の技術を用いて,準同型が高次ホモトピー正規性を持つかどうか調べられる点が強みである.実際,包含写像SU(m) -> SU(n)のp-局所的なホモトピー正規性をいくつかの場合に決定した.このように扱いやすさも実証できており,今後の発展が期待できる.
一様Roe代数のユニタリ群は巨大なホモトピー群を持つ(一様Roe代数のK群と一致)ため難解であるが,実際にホモトピー型を調べる手法を与えた.距離を考慮したトポロジーへの応用も期待できる.

Report

(5 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (16 results)

All 2023 2022 2021 2020 2019 Other

All Journal Article (2 results) (of which Peer Reviewed: 2 results,  Open Access: 1 results) Presentation (10 results) (of which Int'l Joint Research: 4 results,  Invited: 1 results) Remarks (4 results)

  • [Journal Article] Higher homotopy normalities in topological groups2023

    • Author(s)
      Tsutaya Mitsunobu
    • Journal Title

      Journal of Topology

      Volume: 16 Issue: 1 Pages: 234-263

    • DOI

      10.1112/topo.12282

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A short proof for tc(K)=42019

    • Author(s)
      Iwase Norio、Sakai Michihiro、Tsutaya Mitsunobu
    • Journal Title

      Topology and its Applications

      Volume: 264 Pages: 167-174

    • DOI

      10.1016/j.topol.2019.06.014

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Higher homotopy normalities in topological groups2022

    • Author(s)
      Mitsunobu Tsutaya
    • Organizer
      Classifying spaces in homotopy theory: in honour of Ran Levi's 60th Birthday
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Finite propagation operators and Hilbert bundles with end2022

    • Author(s)
      Mitsunobu Tsutaya
    • Organizer
      Topology Seminar, the University of Aberdeen
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Finite propagation operators and Hilbert bundles with end2021

    • Author(s)
      蔦谷充伸
    • Organizer
      東京都立大学幾何学セミナー
    • Related Report
      2021 Research-status Report
  • [Presentation] Homotopy normalities in topological groups2021

    • Author(s)
      蔦谷充伸
    • Organizer
      京都九州信州トポロジー合同セミナー
    • Related Report
      2021 Research-status Report
  • [Presentation] Finite propagation operators and Hilbert bundles with end2021

    • Author(s)
      蔦谷充伸
    • Organizer
      京都大学微分トポロジーセミナー
    • Related Report
      2021 Research-status Report
  • [Presentation] Unstable homotopy types of spaces of finite propagation unitary operators on Z2020

    • Author(s)
      蔦谷充伸
    • Organizer
      関西ゲージ理論セミナー、京都代数トポロジーセミナー合同セミナー
    • Related Report
      2020 Research-status Report
  • [Presentation] Homotopy types of spaces of finite propagation unitary operators on Z2020

    • Author(s)
      Mitsunobu Tsutaya
    • Organizer
      WORKSHOP: unitary operators: spectral and topological properties
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] Homotopy type of the space of finite propagation unitary operators on Z2020

    • Author(s)
      MitsunobuTsutaya
    • Organizer
      Southampton-Kyoto Workshop II
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] Homotopy thoery of An-spaces in Lie groups2019

    • Author(s)
      蔦谷充伸
    • Organizer
      京都大学数学教室談話会
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Characterizations of homotopy fiber inclusion2019

    • Author(s)
      蔦谷充伸
    • Organizer
      ホモトピー論シンポジウム
    • Related Report
      2019 Research-status Report
  • [Remarks] 個人ホームページ

    • URL

      https://www2.math.kyushu-u.ac.jp/~tsutaya/

    • Related Report
      2022 Annual Research Report
  • [Remarks] 個人webサイト

    • URL

      https://www2.math.kyushu-u.ac.jp/~tsutaya/

    • Related Report
      2021 Research-status Report
  • [Remarks] 個人Website

    • URL

      https://www2.math.kyushu-u.ac.jp/~tsutaya/

    • Related Report
      2020 Research-status Report
  • [Remarks] 個人webサイト

    • URL

      https://www3.math.kyushu-u.ac.jp/~tsutaya/

    • Related Report
      2019 Research-status Report

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Published: 2019-04-18   Modified: 2024-01-30  

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