博士, 大阪大学

topology

group cohomology

algebraic topology

homotopy theory

Coxeter group

Artin group

quandle

Wirtinger presentation

crossed module

crossed G-set

Natural Science, Geometry, トポロジー

Natural Science, Algebra, 群論，ホモロジー代数

Bachelor's degree program, School of Science

Master's degree program, Graduate School of Science

Doctoral (PhD) degree program, Graduate School of Science

Associated groups of symmetric quandles
Toshiyuki Akita; Kakeru Shikata
Bulletin of the Polish Academy of Sciences. Mathematics, Institute of Mathematics, Polish Academy of Sciences, 01 May 2026, [Peer-reviewed], [Lead author, Corresponding author]
Scientific journal, A symmetric quandle is a quandle equipped with a good involution. In this paper, we study the structure of the associated groups of symmetric quandles. We examine the relationship between the associated group of a symmetric quandle and that of its underlying quandle. As a consequence, we obtain three distinct structural properties of the associated group of the underlying quandle in terms of that of the symmetric quandle. We also provide a group-theoretic characterization of the associated group of a symmetric quandle and compute its abelianization, which coincides with the first symmetric quandle homology. Furthermore, we express the second quandle homology of a quandle in terms of the associated group of the corresponding symmetric quandle. Finally, we show that a symmetric quandle is embeddable in its associated group if and only if its underlying quandle is.

Groups admitting Wirtinger presentations and Gromov hyperbolic groups
Toshiyuki Akita
Tsukuba Journal of Mathematics (to appear), 2026, [Peer-reviewed]
English, Scientific journal

The second integral homology of even Artin groups
Toshiyuki Akita
Kyushu Journal of Mathematics, (to appear), 2026, [Peer-reviewed]
English, 46301033;45210497

Groups having Wirtinger presentations and the second group homology
Toshiyuki Akita; Sota Takase
Kobe Journal of Mathematics, 41, 33, 39, Nov. 2024, [Peer-reviewed], [Lead author, Corresponding author]
English

Presentations of Schur covers of braid groups
Toshiyuki Akita; Rikako Kawasaki; Takao Satoh
Journal of Group Theory, Walter de Gruyter GmbH, 16 Aug. 2023, [Peer-reviewed], [Lead author]
English, Scientific journal, Abstract

In this paper, we consider several basic facts of Schur covers of the symmetric groups and braid groups.In particular, we give explicit presentations of Schur covers of braid groups.

Embedding Alexander quandles into groups
Toshiyuki Akita
Journal of Knot Theory and Its Ramifications, 32, 02, World Scientific Pub Co Pte Ltd, Feb. 2023, [Peer-reviewed]
Scientific journal, For any twisted conjugation quandle [Formula: see text], and in particular any Alexander quandle, there exists a group [Formula: see text] such that [Formula: see text] is embedded into the conjugation quandle of [Formula: see text]

Structure of the associated groups of quandles
Toshiyuki Akita; Aoi Hasegawa; Masayoshi Tanno
Kodai Mathematical Journal, 45, 2, 270, 281, Tokyo Institute of Technology, Department of Mathematics, 30 Jun. 2022, [Peer-reviewed], [Lead author, Corresponding author]
English, Scientific journal

The adjoint group of a Coxeter quandle
Toshiyuki Akita
Kyoto Journal of Mathematics, 60, 4, 1245, 1260, Duke University Press, 01 Dec. 2020, [Peer-reviewed]
English, Scientific journal

Second mod 2 homology of Artin groups
Toshiyuki Akita; Ye Liu
Algebraic & Geometric Topology, 18, 1, 547, 568, Mathematical Sciences Publishers, 10 Jan. 2018, [Peer-reviewed], [Lead author, Corresponding author]
Scientific journal

Vanishing ranges for the mod p cohomology of alternating subgroups of Coxeter groups
Toshiyuki Akita; Ye Liu
JOURNAL OF ALGEBRA, 473, 132, 141, Mar. 2017, [Peer-reviewed], [Lead author, Corresponding author]
English, Scientific journal

A vanishing theorem for the p-local homology of Coxeter groups
Toshiyuki Akita
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 48, 6, 945, 956, Dec. 2016, [Peer-reviewed]
English, Scientific journal

Periodicity for Mumford-Morita-Miller Classes of Surface Symmetries
Toshiyuki Akita
PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, 47, 4, 897, 909, Dec. 2011, [Peer-reviewed]
English, Scientific journal

Integral Riemann-Roch formulae for cyclic subgroups of mapping class groups
Toshiyuki Akita; Nariya Kawazumi
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 144, 2, 411, 421, Mar. 2008, [Peer-reviewed]
English, Scientific journal

A formula for the Euler characteristics of even dimensional triangulated manifolds
Toshiyuki Akita
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 136, 7, 2571, 2573, 2008, [Peer-reviewed]
English, Scientific journal

On mod p Riemann-Roch formulae for mapping class groups
Toshiyuki Akita
Adv. Stud. Pure Math., 52, 111, 118, Mathematical Society of Japan, 2008, [Peer-reviewed], [Invited]
International conference proceedings

Nilpotency and triviality of mod p Morita-Mumford classes of mapping class groups of surfaces
Toshiyuki Akita
Nagoya Mathematical Journal, 165, 1, 22, Cambridge University Press (CUP), Mar. 2002, [Peer-reviewed]
English, Scientific journal, This paper is concerned with mod p Morita-Mumford classes of the mapping class group Γ_g of a closed oriented surface of genus g ≥ 2, especially triviality and nontriviality of them. It is proved that is nilpotent if n ≡ − 1 (mod p − 1), while the stable mod p Morita-Mumford class is proved to be nontrivial and not nilpotent if n ≢ −1 (mod p − 1). With these results in mind, we conjecture that vanishes whenever n ≡ − 1 (mod p − 1), and obtain a few pieces of supporting evidence.

Periodic surface automorphisms and algebraic independence of Morita-Mumford classes
Toshiyuki Akita; Nariya Kawazumi; Takeshi Uemura
Journal of Pure and Applied Algebra, 160, 1, 1, 11, 08 Jun. 2001, [Peer-reviewed]
English, Scientific journal

Homological infiniteness of Torelli groups
Toshiyuki Akita
Topology, 40, 2, 213, 221, Elsevier BV, Mar. 2001, [Peer-reviewed]
Scientific journal

Homological infiniteness of decorated Torelli groups and Torelli spaces
Toshiyuki Akita
Tohoku Mathematical Journal, 53, 1, 145, 147, Mathematical Institute, Tohoku University, 2001, [Peer-reviewed]
English, Scientific journal, We prove that the rational homology of decorated Torelli groups and Torelli spaces are infinite dimensional when the genus of the reference surface is at least seven, thereby extended one of the main results of [2].[2]T. Akita, Homological infiniteness of Torelli groups, Topology 40 (2001), 213--221.

Cohomology of discrete groups and their finite subgroups
Toshiyuki AKITA
Journal of the Mathematical Society of Japan, 52, 4, 869, 875, Mathematical Society of Japan (Project Euclid), Oct. 2000, [Peer-reviewed]
English, Scientific journal, We investigate the cohomology of a group having finite virtual cohomological dimension in terms of the contributions from finite subgroups. As a result, we prove a variant of Quillen's F-isomorphism theorem which remains valid for an arbitrary commutative ring of coefficients and for suitable families of finite subgroups.

Euler Characteristics of Coxeter Groups, PL-Triangulations of Closed Manifolds, and Cohomology of Subgroups of Artin Groups
Toshiyuki Akita
Journal of the London Mathematical Society, 61, 3, 721, 736, Wiley, Jun. 2000, [Peer-reviewed]
Scientific journal

Aspherical Coxeter Groups That are Quillen Groups
Toshiyuki Akita
Bulletin of the London Mathematical Society, 32, 1, 85, 90, Wiley, Jan. 2000, [Peer-reviewed]
Scientific journal

On the homology of Torelli groups and Torelli spaces
Toshiyuki Akita
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 75, 2, Project Euclid, 01 Feb. 1999, [Peer-reviewed]
Scientific journal

On the Euler characteristic of the orbit space of a proper Γ-complex
Toshiyuki Akita
Osaka J. Math., 36, 4, 783, 791, Osaka University, 1999, [Peer-reviewed]
English

Cohomology and Euler characteristics of Coxeter groups
秋田 利之
Science bulletin of Josai University, Special Issue, 2, 3, 16, 城西大学理学部, 1997, [Invited]
English, Surgery and Geometric Topology : Proceedings of the conference held at Josai University 17-20 September, 1996 / edited by Andrew Ranicki and Masayuki Yamasaki. 本文データは許諾を得てeditorのHPサイトhttp://surgery.matrix.jp/math/josai96/proceedings.html から複製再利用したものである。

On the cohomology of Coxeter groups and their finite parabolic subgroups II
Toshiyuki Akita
Group Representations: Cohomology, Group Actions and Topology, 1, 5, American Mathematical Society, 1997, [Peer-reviewed]

On the Cohomology of Coxeter Groups and Their Finite Parabolic Subgroups
Toshiyuki AKITA
Tokyo Journal of Mathematics, 18, 1, 151, 158, Tokyo Journal of Mathematics, Jun. 1995, [Peer-reviewed]
Scientific journal

Euler characteristics of groups and orbit spaces of free $G$-complexes
Toshiyuki Akita
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 69, 10, Project Euclid, 01 Jan. 1993, [Peer-reviewed]
Scientific journal

カンドルの付随群，Wirtinger表示，群のホモロジー
秋田利之, 2025年度日本数学会秋期総合分科会トポロジー分科会予稿集（特別講演）, Sep. 2025
Japanese

On associated groups of quandles
Toshiyuki Akita, OCAMI Reports Vol. 1 (Quandles and Symmetric Spaces 2023), Mar. 2024, [Lead author, Last author, Corresponding author]
English, Summary national conference

Vanishing theorems for the homology of Coxeter groups and their alternating subgroups (Cohomology theory of finite groups and related topics)
Akita Toshiyuki, RIMS Kokyuroku, 1967, 53, 58, Oct. 2015
Kyoto University, Japanese

Coxeter群のホモロジーの$p$-primary componentについて (変換群の位相幾何と代数構造)
秋田 利之, 数理解析研究所講究録, 1922, 170, 174, Oct. 2014
京都大学, Japanese

Surface symmetries, homology representations, and group cohomology
Akita Toshiyuki, 数理解析研究所講究録, 1581, 103, 108, Feb. 2008
Given a finite group G of automorphisms of a compact Riemann surface, we discuss a relation between Mumford-Morita-Miller classes of odd indices and the homology representation of G. Since most participants were group theorists rather than topologists, I separate the algebraic and the topological ingredients and explain the former in detail.有限群のコホモロジー論の研究 = Cohomology Theory of Finite Groups and Related Topics, 京都大学, English

写像類群の有限部分群と森田-Mumford類（双曲空間及び離散群の研究II）
秋田利之, 数理解析研究所講究録, 1270, 1, 10, Jun. 2002
京都大学, Japanese

群のEuler数について：Coxeter群の場合(代数的組み合わせ論)
秋田 利之, 数理解析研究所講究録, 962, 137, 147, Aug. 1996
京都大学, Japanese

Cohomology of Coxeter groups
Akita Toshiyuki, RIMS Kokyuroku, 838, 98, 109, May 1993
Kyoto University, Japanese

Word labelled oriented graphとWirtinger群の2次ホモロジー
秋田利之
Topological Methods in Geometry and Algebra, 15 Jan. 2026
14 Jan. 2026 - 16 Jan. 2026

Freeness of crossed modules, augmented quandles and augmented racks
秋田利之
カンドルと対称空間2025, 10 Jan. 2026, Invited oral presentation
09 Jan. 2026 - 10 Jan. 2026, [Invited]

Second homology of quotients of braid groups
秋田利之
ホモトピー論シンポジウム2025, 25 Oct. 2025, Japanese
24 Oct. 2025 - 26 Oct. 2025

Artin 群と Coxeter 群のホモロジーの話題から
秋田利之
研究集会「リーマン面に関連する位相幾何学」, 24 Sep. 2025, Japanese, Oral presentation
22 Sep. 2025 - 25 Sep. 2025, [Invited]

カンドルの付随群，Wirtinger表示，群のホモロジー
日本数学会2025年度秋季総合分科会トポロジー分科会, 18 Sep. 2025, Japanese, Invited oral presentation
16 Sep. 2025 - 19 Sep. 2025, [Invited]

Quandles and groups admitting Wirtinger presentations
Toshiyuki Akita
Department of Pure Mathematics, Xi'an Jiaotong-Liverpool University, China, 02 Sep. 2025, Public discourse
[Invited]

（捩れ）Wirtinger表示を持つ群について（その2）
秋田利之
Workshop on mapping class groups and beyond 2025 Summer, 19 Aug. 2025, Oral presentation
17 Aug. 2025 - 20 Aug. 2025

（捩れ）Wirtinger表示を持つ群について（その1）
秋田利之
Workshop on mapping class groups and beyond 2025 Summer, 18 Aug. 2025, Oral presentation
20 Aug. 2025 - Aug. 2025

Wirtinger groups and quandles
秋田利之
One Day Workshop with Rachael Boyd on Algebraic Topology, 29 Jan. 2025, English, Invited oral presentation
[Invited]

群のホモロジーについて
秋田利之
日本数学会北海道支部講演会, 18 Dec. 2024
18 Dec. 2024 - 18 Dec. 2024, [Invited]

Groups having Wirtinger presentations and the second group homology
秋田利之
第50回変換群論シンポジウム, 23 Nov. 2024
22 Nov. 2024 - 22 Nov. 2024, [Invited]

代数トポロジーへの誘い
秋田利之
数学なんでもセミナー（公立千歳科学技術大学）, 04 Oct. 2024
[Invited]

Groups having Wirtinger presentations and group homology
秋田利之
研究集会「トポロジーとコンピュータ 2024」, 18 Sep. 2024, Invited oral presentation
17 Sep. 2024 - 19 Sep. 2024, [Invited]

Freeness of crossed modules, augmented quandles, and crossed G-sets
秋田利之
研究集会「曲面の写像類群と群の不変量」, 14 Aug. 2024
20 Aug. 2024, [Invited]

Free crossed modules and free augmented quandles
秋田利之
ホモトピー論シンポジウム2024, 29 Jun. 2024, Oral presentation
28 Jun. 2024 - 30 Jun. 2024

カンドルの付随群について
秋田利之
研究集会「カンドルと対称空間」, 26 Jan. 2024
26 Jan. 2024 - 27 Jan. 2024, [Invited]

トポロジー入門（オイラー数の話）
秋田利之
令和5年度北海道札幌北高等学校数学ゼミナール, 04 Dec. 2023, Japanese
[Invited]

Alexanderカンドルの共役カンドルへの埋め込み
秋田利之
日本数学会秋季総合分科会, 21 Sep. 2023
20 Sep. 2023 - 23 Sep. 2023

カンドルのassociated groupについて
秋田利之; 長谷川蒼
日本数学会秋季総合分科会, 15 Sep. 2022, Japanese, Oral presentation
13 Sep. 2022 - 16 Sep. 2022

ブレイド群の中心拡大
秋田利之
代数的位相幾何学の軌跡と展望, 11 Mar. 2022
[Invited]

The adjoint group of a Coxeter quandle
Toshiyuki Akita
日本数学会2021年度年会, 15 Mar. 2021, Japanese, Oral presentation
15 Mar. 2021 - 18 Mar. 2021

Artin群とCoxeterカンドルの随伴群のコホモロジー
秋田利之
森本雅治先生退職記念研究集会, 15 Feb. 2020, Invited oral presentation
15 Feb. 2020 - 16 Feb. 2020, [Invited]

On the cohomology of Coxeter groups and related groups
Toshiyuki Akita
The Third Pan Pacific International Conference on Topology and Applications (PPICTA), 11 Nov. 2019, English, Oral presentation
成都(中国), [International presentation]

Central extensions of Coxeter groups and Artin groups
Toshiyuki Akita
Branched Coverings, Degenerations, and Related Topics 2019, 05 Mar. 2019
広島大学(東広島キャンパス)大学院理学研究科, [Invited]

Coxeter群とArtin群のコホモロジーについて
秋田利之
九州大学トポロジー金曜セミナー, 14 Dec. 2018, Invited oral presentation
[Invited]

Coxeterカンドルの随伴群
秋田利之
九州大学数理談話会, 10 Dec. 2018, Japanese, Invited oral presentation
[Invited], [Domestic Conference]

Coxeterカンドルとルート系の随伴群
秋田利之
2018年度ホモトピー論シンポジウム, 02 Nov. 2018, Japanese, Invited oral presentation
[Invited], [Domestic Conference]

Coxeterカンドルとルート系の随伴群
秋田利之
ホモトピー沖縄, 20 Sep. 2018, Japanese, Invited oral presentation
[Invited], [Domestic Conference]

Coxeter groups, Coxeter quandles, and Artin groups
秋田利之
Matroids, reflection groups, and free hyperplane arrangements, 15 Jun. 2018, English, Invited oral presentation
RIMS, [Invited], [International presentation]

Coxeter groups, Artin groups and Coxeter quandles
Toshiyuki Akita
研究集会「ストリングトポロジーとその周辺」, 09 Dec. 2017, Japanese, Invited oral presentation
四季の湯強羅静雲荘, [Invited], [Domestic Conference]

カンドルと対称群の中心拡大
秋田利之
北海道大学数学教室談話会, 05 Oct. 2017, Japanese, Invited oral presentation
[Invited]

On the mod p cohomology of Coxeter groups and their alternating subgroups
Toshiyuki Akita
ホモトピー論シンポジウム, 13 Nov. 2016, English, Oral presentation
県立広島大学サテライトキャンパス, [Domestic Conference]

Second mod 2 homology of Artin groups
Toshiyuki Akita
東大火曜トポロジーセミナー, 08 Nov. 2016, Japanese, Public discourse
[Invited], [Domestic Conference]

Cohomology of Coxeter groups and related groups
Toshiyuki Akita
Perspectives on arrangements and configuration spaces, 09 Sep. 2016, English, Invited oral presentation
Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore di Pisa, [Invited], [International presentation]

Crossed modules and Artin groups
Toshiyuki Akita
京大代数トポロジーセミナー, 02 Jun. 2016, Japanese, Public discourse
[Invited], [Domestic Conference]

幾何学講義, 2024年, 修士課程, 理学院

基礎数学演習Ｂ１, 2024年, 学士課程, 理学部

基礎数学Ｂ１, 2024年, 学士課程, 理学部

幾何学続論, 2024年, 学士課程, 理学部

線形代数学Ⅱ, 2024年, 学士課程, 全学教育

基礎数学演習Ｂ, 2024年, 学士課程, 理学部

基礎数学Ｂ, 2024年, 学士課程, 理学部

科学・技術の世界, 2024年, 学士課程, 全学教育

Dec. 1990 - Present
日本数学会

群のホモロジー、Wirtinger群、カンドル、クロス加群の位相幾何学的研究
科学研究費助成事業
Apr. 2024 - Mar. 2028
秋田 利之
日本学術振興会, 基盤研究(C), 北海道大学, 24K06727

Cohomology of Coxeter groups, Artin groups, and Coxeter quandles
Grants-in-Aid for Scientific Research
01 Apr. 2020 - 31 Mar. 2024
秋田 利之; 吉永 正彦
（1）カンドルQの随伴群Ad(Q)はカンドルの構造の解明において最も重要な群であるが、表示で定義されるため調べるのが難しい。Grana-Heckenberger-Vendramin（2011）は有限カンドルQに対しAd(Q)の有限商群F(Q)を導入することにより様々な結果を得ていた。本研究では丹野信義、長谷川蒼と共にGrana等の構成を無限カンドルを含める形で一般化し（i）Ad(Q)の中心拡大としての特徴付け（ii）Ad(Q)の交換子群の構造（iii）Ad(Q)の分類空間のホモトピー・プルバックとしての特徴付けなど多くの結果を得た。結果を纏めた論文はKodai Mathematical Journalに掲載が決定している。
（2）群のSchur被覆はJ. Schur（1911）による有限群の射影表現の研究において見出された概念であり、完全群の普遍中心拡大の一般化となっている。一方、クロス加群（crossed module）はJ. H. C. Whitehead（1949）による低次のホモトピー群の研究において導入された概念であり、Postnikov不変量を介して群の3次コホモロジー群と関係している。クロス加群はホモトピー2型（homotopy 2-type）のモデルや群の高次元化とみなせることから、ホモトピー論を超えて様々な数学と関連している。さらにクロス加群から本研究の主な対象であるカンドルが誘導される。Huebschmann（2012）はブレイド群のSchur被覆がブレイド群上の一元生成自由クロス加群であることを示している。そこで本研究では研究代表者の学生であった川崎理佳子と共同でブレイド群のSchur被覆の有限表示を求めた。表示を求める際には対称群のZ/2に値を持つ2コサイクルの具体的な値の計算が鍵となった。
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 20K03600

Topological studies on cohomology of Artin groups and related topics
Grants-in-Aid for Scientific Research
01 Apr. 2017 - 31 Mar. 2020
Akita Toshiyuki
The adjoint group of a Coxeter quandle is an intermediate group between the corresponding Coxeter group and the Artin group. The study of cohomology of adjoint groups is important for the study of cohomology of Coxeter groups and Artin groups. As results, we determined the rational cohomology rings of all adjoint groups, proved that Hepworth families of adjoint groups have homology stability, and evaluated vanishing ranges of mod p cohomology groups of adjoint groups.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 17K05237

Cohomology of mapping class groups, Coxeter groups and Artin groups
Grants-in-Aid for Scientific Research
01 Apr. 2014 - 31 Mar. 2017
Akita Toshiyuki; SATOH Takao
We studied group cohomology of Coxeter groups, Artin groups and related groups. As for Coxeter groups, we obtain (1) a vanishing theorem for the p-local homology of Coxeter groups (2) a vanishing theorem for the mod p cohomology of alternating subgroups of finite Coxeter groups. As for Artin groups (3) we determined the second mod 2 homology of arbitrary Artin groups. (2) and (3) are joint works with Ye Liu. Finally, we proved that the adjoint group of an arbitrary Coxeter quandle is both a central extension of a Coxeter group W by a free abelian group and a semi-direct product of the commutator subgroup of a Coxeter group W and a free abelian group.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 26400077

Constructions of cocycles of finite groups by characteristic classes
Grants-in-Aid for Scientific Research
2011 - 2013
AKITA Toshiyuki; HASHIMOTO Yoshitake
For a finite Galois covering of a compact Riemann surface with a monodromy group G, we constructed cocycles representing the characteristic classes (Mumford-Morita-Miller classes) associated with a Galois covering, by using transfer homomorphisms in group cohomology and Kawazumi-Uemura formula. In addition, we construct mod p cocycles for such classes by using "periodicity phenomena" and Steenrod operations. Moreover, we prove a vanishing theorem for p-local homology of Coxeter groups. The key ingredient was equivariant homology of Coxeter complexes.
Japan Society for the Promotion of Science, Grant-in-Aid for Challenging Exploratory Research, Hokkaido University, 23654018

Coho mology of discrete groups and characteristic classes of flat bundles
Grants-in-Aid for Scientific Research
2008 - 2011
AKITA Toshiyuki; OHMOTO Toru; WATANABE Tadayuki; YOSHIDA Tomoyuki; KURIBAYASHI Katsuhiko; YAGITA Nobuaki
We construct a functorial framework to deal with all finite transformation groups on closed surfaces (Galois covers on closed surfaces) simultaneously. Characteristic classes associated with such transformation groups, Mumford-Morita-Miller classes and Newton classes, are shown to be natural transformations of functors. These characteristic classes are shown to satisfy Riemann-Roch type formulae.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, 20340008

Geometry of Groups and Moduli Spaces (2)
Grants-in-Aid for Scientific Research
2007 - 2009
MORITA Shigeyuki; FURUTA Mikio; TSUBOI Takashi; KOHNO Toshitake; KAWAZUMI Nariya; MATSUMOTO Makoto; MITSUMATSU Yoshihiko; KITANO Teruaki; FUJIWARA Koji; MURAKAMI Jun; AKITA Toshiyuki; HIROSE Susumu; MORIFUJI Takayuki; SUZUKI Masaaki; KASAHARA Yasushi; SAKASAI Takuya; DIETER Kotschick; ROBERT Penner
The moduli space of Riemann surfaces and the moduli space of graphs, as well as their associated modular groups such as the mapping class groups of surfaces and the automorphism groups of free groups, are among the most important research subjects of diverse branches of mathematics including algebraic geometry, complex analysis, differential geometry, topology and mathematical physics. The present project investigated these moduli spaces and modular groups, mainly form the point of view of topology, and obtained many interesting results. Furthermore, we obtained new results as well as conjectures in the closely related theories of 3 and 4 dimensional manifolds and transversely symplectic foliations. We also proposed a deep problem towards new directions of our research including number theory.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), The University of Tokyo, 19204003

Topological Studies around Riemann Surfaces
Grants-in-Aid for Scientific Research
2006 - 2009
KAWAZUMI Nariya; MATSUMOTO Yukio; MORITA Shigeyuki; HASHIMOTO Yoshitake; SHIBUKAWA Youichi; AKITA Toshiyuki; ENDO Hisaaki; ASUKE Taro; TADOKORO Yuuki
Bene, Penner and the principal investigator discovered fatgraph Magnus expansions, which connects a combinatorial structure of a Riemann surface directly to some algebraic aspects of the mapping class groups. The principal investigator also discovered a new analytic invariant of a closed Riemann surface to describe how curved the moduli space of Riemann surfaces is. Kuno and the principal investigator discovered a new connection between two refinements of the intersection form on a Riemann surfaces, the Goldman Lie algebras and the Lie algebras of symplectic derivations. As an application, they proved a non-commutative analogue of the Picard-Lefschetz formula.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), The University of Tokyo, 18204002

Equivariant theory of Chern classes, Integrals based on Euler characteristics and related topics
Grants-in-Aid for Scientific Research
2005 - 2008
OHMOTO Toru; SHOJI Yokura; TATSUO Suwa; GO-O Ishikawa; TOSHIYUKI Akita
研究代表者の同変特異チャーン類理論を基礎に, 種々のオビフォルド・特異チャーン類を定義した. 特に, 古典的群論における置換表現の数え上げ公式をオビフォルド特性類に拡張したものとして, 代数多様体の対称積に関するオビフォルド・特異チャーン類の生成母関数公式を示した. これは, 特異チャーン類理論の数え上げ幾何あるいはマッカイ対応への応用に向けた足場となる.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, 17340013

Studies on the Cohomology of Mapping Class Groups, Coxeter Groups and Artin Groups
Grants-in-Aid for Scientific Research
2005 - 2007
AKITA Toshiyuki; IZEKI Hiroyasu; HIROSE Susumu; HOSAKA Tetsuya; KAWAZUMI Nariya; OHMOTO Toru
The cohomology of discrete groups, such as mapping class groups of closed surfaces, Coxeter groups and Artin groups, is one of the important objects in topology as well as geometry In this research project, we studied the cohomology of discrete groups. The following three subjects were emphasized in the project:
(1) Relations with the cohomology of finite subgroups
(2) Actions of discrete groups on manifolds/complexes and combinatorial structures
(3) Algebraic methods (such as combinatorics and free resolutions)
Concerning of (1), Akita and Kawazumi proved integral Riemann-Roch formulae for cyclic subgroups of mapping class groups, which are variants of Grothedieck-Riemann-Roch theorem for integral cohomology. Concerning of (2), Izeki and Hosaka obtained various results concerning of group actions and geometric structures Finally, concering of (3), Akita proved alternative formulae for the Euler characteristics of even dimensional triangulated manifolds. The key ingredient to prove the formulae is generalized Dehn-Sommerville equations obtained by Klee. In addition, Akita showed that mod p Riemann-Roch formulae hold for various cases, by using Kummer's congruences in classical number theory.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 17540056

Applied singularity theory to exterior differential systems
Grants-in-Aid for Scientific Research
2002 - 2005
ISHIKAWA Goo; YAMAGUCHI Keizo; IZUMIYA Shyuichi; ONO Kaoru; AKITA Toshiyuki; OHMOTO Toru
The following results are obtained : Classification of generic singularities in geometric solutions to Monge-Ampere equations. Clarification of Goursat-Legendre correspondence. Basic investigations on singular Legendre submanifolds, in particular singular Legendre curves. Bifurcation of singularities of developables. Estimate on doubly degenerate submanifolds under projective duality. Introduction of the notion of singular coisotropic mappings. Proof of the localization theorem on the symplectic moduli spaces. Basic study on mapping space quotients. Classification theory of singular Legendre knots. Discovery of new singularities of solutions to Monge-Ampere equations in dimension three. Classification of uni-modal curve singularities and determination of their symplectic moduli spaces. Discovery of remarkable similarity between diffeomorphism classification of plane curve singularities and contactomorphism classification of associated Legendre curve singularities.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), HOKKAIDO UNIVERSITY, 14340020

写像類群のコホモロジーと整係数Riemann-Roch公式の位相幾何学的研究
科学研究費助成事業
2002 - 2004
秋田 利之
写像類群に対するGrothendieck-Riemann-Rochの定理の、整係数コホモロジーにおける類似(整係数Grothendieck-Riemann-Roch公式)を証明すること、その応用として写像類群の整係数コホモロジーの構造を解明することを目的として研究を進めた。今年度は(i)コホモロジー作用素とGysin準同型(ファイバーに沿った積分)の関係(ii)前項の結果の整係数Grothendieck-Riemann-Roch公式への応用を中心に研究を進め以下の結果を得た。
1.向き付けられた閉多様体をファイバーとするファイバー束に対し、コホモロジー作用素(Steenrod作用素)とGysin準同型との非可換性が、相対接束(ファイバーに沿った接束)の全Stiefel-Whitney類または全Wu類で記述されることを前年度に示したが、この結果の別証明をBecker-Gottliebトランスファーを用いて与えた。
2.前項の結果と整数論のKummerの恒等式を組み合わせることにより、写像類群に対する整係数Grothendieck-Riemann-Roch公式の素数pを法とする還元(mod p Grothendieck-Riemann-Roch公式)が無限に多くの場合に正しいことを証明した。
3.写像類群の安定森田-Mumford類には非自明な関係式がないことが知られていたが、そのmod p還元には、多くの非自明な関係式があることを示した。
日本学術振興会, 若手研究(B), 北海道大学, 14740032

Complex analytic approach towards topology studies on the mapping class ganups for surfaces
Grants-in-Aid for Scientific Research
2002 - 2004
KAWAZUMI Nariya; MATSUMOTO Yukio; MORITA Shigeyuki; HASHIMOTO Yoshitake; SHIBUKAWA Youichi; AKITA Toshiyuki
We discovered a close relation between Stasheff associahedrons and (generalized) Magnus expansions of a free group. A certain part of the twisted Morita-Mumford classes can be extended to the automorphism group of a free group. It is parametrized by Stasheff associahedrons "infinitesimally" and "combinatorially" how the extended Johnson maps are far from true group homomorphisms.
We extended our theory on harmonic Magnus expansions to the universal family of Riemann surfaces. This yields another series of canonical 1 forms on the universal family than what we have already obtained on the moduli space. As a corollary, we obtained a proof that the first Jonson map and the (0,3)-twisted Morita-Mumford class coincides with each other as differential forms on the moduli space.
The Magus representation of the automorphism group of a free group was constructed in an intrinsic manner. Here 'intrinsic' means 'with no use of Fox' free differentials.'
We co-organized a workshop entitled "Toward the future of the topological study of manifolds" in November 2004.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), The University of Tokyo, 14540065

Mapping Class Group of Surfaces and Geometry of Moduli Spaces
Grants-in-Aid for Scientific Research
2001 - 2003
MORITA Shigeyuki; NAKAMURA Hiroaki; KAWAZUMI Nariya; FURUTA Mikio; MURAKAMI Jun; AKITA Toshiyuki
In this project, we focussed on the study of the structure of the mapping class group of surfaces (m.c.g. for short) as well as the moduli space of compact Riemann surfaces, together with various problems closely related with this. They include the following thema : cohomology group of m.c.g., the theory of the Floer homotopy types, topological invariants based on gauge theory, construction of the harmonic Magnus expansion of m.c.g., structure of the Grothendieck-Teichm＼"uller group, the volume conjecture, non-commutative geometry in dimensions 3,4, finite subgroups of m.c.g., the Jones representation of m.c.g., relation between m.c.g. with 4-dimensional topology. From the interactions of these thema, we found new directions of research such as the relation between the geometry of m.c.g. and the symplectic topology as well as the comparaison between m.c.g. and the outer automorphism group of free groups.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), The University of Tokyo, 13440017

写像類群のコホモロジーと曲面束の特性類の位相幾何学的研究
科学研究費助成事業
2000 - 2001
秋田 利之
本年度は主に向きづけられた閉曲面の写像類群の森田-Mumford類の素数pを法とする還元の自明性、写像類群の有限部分群の森田-Mumford類と2次元同変ボルディズム群およびG-符号数との関係、同変コホモロジーの局所化定理と森田-Mumford類との関係の三つの課題を中心に研究を進めた。それぞれの課題について得られた成果を項目にわけて以下に述べる。
第1に素数pに対し、写像類群の部分群Gが閉曲面の余接束のmod pコホモロジーに自明に作用するならば、Gの森田-Mumford類はすべて自明であることを示した。とくにスピン写像類群のmod2森田-Mumford類は全て自明であることを証明した。
第2に昨年度に引き続き、2次元(有向)同変ボルディズム群と写像類群の有限部分群の森田-Mumford類との関係を研究した。まず有限群の2次元同変ボルディズム群から有限群の分類空間のコホモロジー群への準同型を導入し、その準同型を用いて奇数次の森田-Mumford類が記述できることを示した。さらにその準同型とG-符号数との関係をEichlerの跡公式などを用いて調べることにより、奇数次の森田-Mumford類の2倍がG-符号数で決まることの簡単な証明を得た。
第3に同変コホモロジーの局所化定理を用いて写像類群の有限部分群の森田-Mumford類の不動点公式(植村-河澄公式)の別証明を得た。さらに同変K理論の局所化定理を用いてコホモロジー表現のChern類との関係を調べた。
日本学術振興会, 奨励研究(A), 北海道大学, 12740030

A topological study of the moduli space of Riemann surface
Grants-in-Aid for Scientific Research
1999 - 2001
KAWAZUMI Nariya; OHBA Kiyoshi; MORITA Shigeyuki; MATSUMOTO Yukio; AKITA Toshiyuki; SHIBUKAWA Youichi
Rational cohomology of the mapping class group (Morita and Kawazumi) : We proved the Morita-Mumford classes generate the primary approximation to the cohomology of the moduli of Riemann surfaces induced by the Johnson homomorphisms even in the unstable range. We gave a complete description how the corresponding twisted Morita-Mumford classes behave when a finite graph degenerates.
Differential geometry of the moduli and Magnus expansions (Kawazumi) : Using Magnus expansions of a free group, we obtain an alternative proof of the IH-relation among the Johnson homomorphims. The notion of the harmonic Magnus expansion, which is canonicaly given by a complex structure of a surface, gives an interpretation of differential forms representing Morita-Mumford classes. We compute the quasi-conformal variation of the harmonic Magnus expansions as an explicit quadratic differential. We expect that the quadratic differential would give the key to the Johnson images of the mapping class groups.
Torsion cohomology of the mapping class groups (Akita and Kawazumi) : Akita has given fascinating conjectures related to Morita-Mumford classes on the whole mapping class groups. We proved them for any semi-free cyclic subgroup. Kawazumi proved them for the hyperelliptic mapping class groups. Akita proved the twice of the odd Morita-Mumford classes are functions of G-signatures for any finite subgroup G of the mapping class groups.
Bruschi-Calogero equation (Shibukawa and Kawazumi) : We gave all the meromorphic solutions of the equation. Shibukawa gave the complete classification of the R-matrices acting on the germs of meromorphic functions.
The details and other results are reported in the official booklet written in Japanese.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), The University of Tokyo, 11640054

反復的曲面束の位相幾何学的研究
科学研究費助成事業
1997 - 1998
秋田 利之
本年度は主に向きづけられた閉曲面の周期的微分同相写像と曲面束の特性類との関係、および曲面束のmod 2特性類(とくにmod 2森田-Mumfrod類)の構造について研究した。
1. 閉曲面の周期的微分同相に対し、その写像トーラスのη不変量、G-符号数、および第一森田-Mumford類の関係を明らかにした。とくに写像トーラスのη不変量のG-符号数による表示を見い出し、また閉曲面の周期的微分同相(あるいは写像類群の有限部分群)の第一森田-Mumford類の消滅が写像トーラスのη不変量の整数性で特徴づけられることを証明した。さらに閉曲面の周期的微分同相の奇数次の森田-Mumford類がG-符号数とLefschetz数で決定されることを発見した。一方で偶数次の森田-Mumford類はG-符号数とLefschetz数のみでは決らないことを示した。
2. 写像類群のmod 2森田-Mumford類の消滅について種々の結果を得た。とくに(1)種数が2または3の写像類群(2)写像類群の有限部分群(3)レベル2の写像類群に対してはそれらのmod 2森田-Mumford類が消滅することを証明した。また一般にmod 2森田-Mumford類が幕零であることを証明した。これらの結果を用いて曲面束の同境に関して種々の結果を得た。とくに種数2の曲面束あるいは自明な同伴Hodge束をもつ曲面束の全空間が有向零同境であることを証明した。
3. 種数7以上の有向曲面のTorelli群の有理コホモロジーが穴(puncture)と境界成分の個数によらず常に無限次元であることを証明した。これは昨年度に得られた結果を拡張したものである。
日本学術振興会, 奨励研究(A), 福岡大学, 09740072

On the construction and classification of the finite geometry
Grants-in-Aid for Scientific Research
1997 - 1998
ODA Nobuyuki; AKIYAMA Kenji; AKITA Toshiyuki; KUROSE Takashi; INOUE Atsushi
Geometrical constructions in homotopy sets were studied. We obtained results on the GAMMA-Whitehead product and the GAMMA-Hopf construction. We introduced the transformation between pairings and copairings and showed its applications. We obtained a formula for the smash product. We obtained a generalization of the Hardie-Jansen product and studied its properties. Dual results are also studied.
For geometrical construction in operator algebras, Tomita-Takesaki theory was studied. We obtained results on unbounded C^*seminorms on *-algebra and standard weights which enable us to develop unbounded Tomita-Takesaki theory.
We constructed explicit examples of surfaces in affine spaces of dimension three and four. We gave a necessary and sufficient condition on surfaces in a three-dimensional affine space to be metric when the surfaces have non-zero constant Gauss-Kronecker curvature.
The cohomology of mapping class groups was studied. We obtained a relation among periodic automorphisms of closed surfaces and the eta-invariant of their mapping tori. We also obtained various vanishing theorems of mod 2 Morita-Mumford classes.
The Schur ring of product type was characterized by the existence of a subgroup of a collineation group. The existence of a Schur ring of produt difference set type is characterized by a finite projective plane of order n with a collineation group of order n(n - 1).
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Fukuoka University, 09640306

Homotoy Theory of Classifying Spaces
Grants-in-Aid for Scientific Research
1997 - 1998
ISHIGURO Kenshi; AKITA Toshiyuki; KUROSE Takashi; ODA Nobuyuki
The research on the classifying spaces of compact Lie groups has been one of the major area in Homotoy Theory. Our results obtained during 1997 through 1998 are basically concerned with maps between classifying spaces and their applications. Dwyer-Wilkerson defined ap-compact group and studied its properties. The purely homotopy theoritic object appears to be a good generalization of a compact Lie group. A p-compact group has rich structure, such as a maximal torus, a Weyl group, etc. A note wrtten by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of the classifying spaces of p-compact groups are being made.
We state here our main results. First, we consider the maps of p-compact groups of the form BX * BY*BZ.The main theorem shows that if the restriction map on BY is a weak epimorphism, then the restriction on BX should factor through the classifying spaces of the center of the p-compact group Z.Next, for G =S^3 * .. * S^3, let X be a genus of BG.We investigate the monoid of rational equivalences of X, denoted by epsilon(X). It is shown that a submonoid of epsilon_0(X), denoted by delta_00(X), determines the decomposability of the space X.We also show converses to some known results for the classifying spaces of p-toral groups or p-compact toral group. Suppose G is a compact Lie group. The following results are obtained. If there is a positive integer k such that the n-th homotopy groups of the p-completion of BG are zero for all n k then the loop space of this space is a p-compact toral group. If the canonical map Rep(G, K)*[BG, BK] is bijective for any compact connected Lie group K, then G is a p-toral group. in addition, our work containesa research on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Fukuoka University, 09640138

正曲率及びワイル共形曲率と多様体の空間
科学研究費助成事業
1996 - 1996
陶山 芳彦; 荻 秀和; 秋田 利之; 高倉 樹; 黒瀬 俊; 吉田 守
1.どのような共形平坦な多様体が,定曲率空間の超曲面として実現されるかという問題を研究し,4次元以上の(ある種の)共形平坦な多様体に関して,それらの多様体から定曲率空間への共形的はめ込みの具体的構成法を発見した。更に,上の構成ではめ込み可能な多様体の共形類を決定するために,それらの超曲面から球面への展開写像の構成を行った。
2.射影平坦で捩れをもたないアフィン接続が与えられた単連結多様体の射影展開写像について研究し,次ぎの結果を得た。3次元以上で接続に関して極を持つ多様体のリッチ曲率が対称で負定値ならば,その展開写像は単射であり,像は射影空間の凸集合となる。
3.シンプレクティック・トーリック多様体上に,退化した不変偏極の族を構成し,同伴する幾何学的量子化の推移において,特殊なラグランジュ部分多様体上への極所化現象が起こることを示した。
4.閉曲面上の平坦接続のモジュライ空間の幾何学的量子化(一般化されたテ-タ関数の空間)の次元に関するフェアリンデの分解公式の,シンプレクティック幾何的な証明について研究した。その方法は,ハミルトン的群作用の下での幾何学的量子化に関する重複度公式を応用するというものである。
日本学術振興会, 基盤研究(C), 福岡大学, 08640146