Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

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Profile and Settings

Profile and Settings

• Name (Japanese)

  Akita

• Name (Kana)

  Toshiyuki

• Name

  200901097733036240

Achievement

Research Interests

• group cohomology   quandle   homotopy theory   algebraic topology   Artin group   classifying space   topology   Coxeter group   Euler characteristic   mapping class group   

Research Areas

• Natural sciences / Geometry

Research Experience

• 2017/04 - Today Hokkaido University Faculty of Science
• 2007/04 - 2017/03 Hokkaido University Faculty of Science
• 2006/04 - 2007/03 Hokkaido University Faculty of Science
• 1999/08 - 2006/03 Hokkaido University
• 1995/04 - 1999/07 Fukuoka University Faculty of Science
• 1994/04 - 1995/03 日本学術振興会 特別研究員(DC3)

Published Papers

• Groups having Wirtinger presentations and the second group homology

  Toshiyuki Akita, Sota Takase

  Kobe Journal of Mathematics 41 33 - 39 2024/11 [Refereed]
  
• Presentations of Schur covers of braid groups

  Toshiyuki Akita, Rikako Kawasaki, Takao Satoh

  Journal of Group Theory 1433-5883 2023/08/16 [Refereed]
   

  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) 0218-2165 2023/02 [Refereed]
   

  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 0386-5991 2022/06/30 [Refereed]
  
• The adjoint group of a Coxeter quandle

  Toshiyuki Akita

  Kyoto Journal of Mathematics 60 (4) 1245 - 1260 2156-2261 2020/12/01 [Refereed]
  
• Second mod 2 homology of Artin groups

  Toshiyuki Akita, Ye Liu

  Algebraic & Geometric Topology 18 (1) 547 - 568 1472-2747 2018/01/10 [Refereed]
  
• Vanishing ranges for the mod p cohomology of alternating subgroups of Coxeter groups

  Toshiyuki Akita, Ye Liu

  JOURNAL OF ALGEBRA 473 132 - 141 0021-8693 2017/03 [Refereed][Not invited]
   

  We obtain vanishing ranges for the mod p cohomology of alternating subgroups of finite p-free Coxeter groups. Here a Coxeter group W is p-free if the order of the product st is prime to p for every pair of Coxeter generators s, t of W. Our result generalizes those for alternating groups formerly proved by Kleshchev-Nakano and Burichenko. As a byproduct, we obtain vanishing ranges for the twisted cohomology of finite p-free Coxeter groups with coefficients in the sign representations. In addition, a weak version of the main result is proved for a certain class of infinite Coxeter groups. (C) 2016 Elsevier Inc. All rights reserved.
• A vanishing theorem for the p-local homology of Coxeter groups

  Toshiyuki Akita

  BULLETIN OF THE LONDON MATHEMATICAL SOCIETY 48 (6) 945 - 956 0024-6093 2016/12 [Refereed][Not invited]
   

  Given an odd prime number p and a Coxeter group W such that the order of the product st is prime to p for all Coxeter generators s, t of W, we prove that the p-local homology groups H-k(W, Z((p))) vanish for 1 <= k <= 2(p - 2). This generalizes a known vanishing result for symmetric groups due to Minoru Nakaoka.
• Periodicity for Mumford-Morita-Miller Classes of Surface Symmetries

  Toshiyuki Akita

  PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 47 (4) 897 - 909 0034-5318 2011/12 [Refereed][Not invited]
   

  We prove periodicity for mod p Mumford-Morita-Miller classes of surface symmetries and thereby for finite subgroups of mapping class groups. As an application, we obtain a couple of vanishing results for mod p Mumford-Morita-Miller classes for surface symmetries.
• 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 0305-0041 2008/03 [Refereed][Not invited]
   

  The first author conjectured certain relations for Morita-Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann-Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.
• A formula for the Euler characteristics of even dimensional triangulated manifolds

  Toshiyuki Akita

  PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 136 (7) 2571 - 2573 0002-9939 2008 [Refereed][Not invited]
   

  An alternative formula for the Euler characteristics of even dimensional triangulated manifolds is deduced from the generalized Dehn-Sommerville equations.
• On mod p Riemann-Roch formulae for mapping class groups

  Toshiyuki Akita

  Adv. Stud. Pure Math. 52 111 - 118 2008 [Refereed][Invited]
  
• Nilpotency and triviality of mod p Morita-Mumford classes of mapping class groups of surfaces

  Toshiyuki Akita

  Nagoya Mathematical Journal 165 1 - 22 0027-7630 2002/03 [Refereed]
   

  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 0022-4049 2001/06/08 [Refereed][Not invited]
   

  We prove a vanishing theorem for the Morita-Mumford classes on periodic surface automorphisms, and construct enough periodic automorphisms to give an alternative and elementary proof of the stable rational algebraic independence of the Morita-Mumford classes, originally shown by Miller (J. Differential Geom. 24 (1986) 1-14) and Morita (Invent. Math. 90 (1987) 551-557). © 2001 Elsevier Science B.V.
• Homological infiniteness of Torelli groups

  Toshiyuki Akita

  Topology 40 (2) 213 - 221 0040-9383 2001/03 [Refereed]
  
• Homological infiniteness of decorated Torelli groups and Torelli spaces

  Toshiyuki Akita

  Tohoku Mathematical Journal 東北大学 53 (1) 145 - 147 0040-8735 2001 [Refereed]
   

  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 The Mathematical Society of Japan 52 (4) 869 - 875 0025-5645 2000/10 [Refereed]
   

  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 0024-6107 2000/06 [Refereed]
  
• Aspherical Coxeter Groups That are Quillen Groups

  Toshiyuki Akita

  Bulletin of the London Mathematical Society 32 (1) 85 - 90 0024-6093 2000/01 [Refereed]
  
• On the homology of Torelli groups and Torelli spaces

  Toshiyuki Akita

  Proceedings of the Japan Academy, Series A, Mathematical Sciences 75 (2) 0386-2194 1999/02/01 [Refereed]
  
• On the Euler characteristic of the orbit space of a proper Γ-complex

  Toshiyuki Akita

  Osaka J. Math. 大阪大学 36 (4) 783 - 791 0030-6126 1999 [Refereed]
  
• Cohomology and Euler characteristics of Coxeter groups

  秋田 利之

  Science bulletin of Josai University, Special Issue 城西大学理学部 2 3 - 16 1342-9590 1997 [Not refereed][Invited]
   

  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 0082-0717 1997 [Refereed]
  
• On the Cohomology of Coxeter Groups and Their Finite Parabolic Subgroups

  Toshiyuki AKITA

  Tokyo Journal of Mathematics 18 (1) 151 - 158 0387-3870 1995/06 [Refereed]
  
• Euler characteristics of groups and orbit spaces of free $G$-complexes

  Toshiyuki Akita

  Proceedings of the Japan Academy, Series A, Mathematical Sciences 69 (10) 0386-2194 1993/01/01 [Refereed]
  

MISC

• 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  2015/10
• 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  2014/10
• Surface symmetries, homology representations, and group cohomology

  Akita Toshiyuki  数理解析研究所講究録  1581-  103  -108  2008/02  [Not refereed][Not invited]
   

  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
• 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  2002/06  [Not refereed][Not invited]
  
• 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  1996/08
• Cohomology of Coxeter groups

  Akita Toshiyuki  RIMS Kokyuroku  838-  98  -109  1993/05

Presentations

• Groups having Wirtinger presentations and the second group homology  [Invited]

  秋田利之

  第50回変換群論シンポジウム  2024/11
• 代数トポロジーへの誘い  [Invited]

  秋田利之

  数学なんでもセミナー（公立千歳科学技術大学）  2024/10
• Groups having Wirtinger presentations and group homology  [Invited]

  秋田利之

  研究集会「トポロジーとコンピュータ 2024」  2024/09
• Freeness of crossed modules, augmented quandles, and crossed G-sets  [Invited]

  秋田利之

  研究集会「曲面の写像類群と群の不変量」  2024/08
• Free crossed modules and free augmented quandles  [Not invited]

  秋田利之

  ホモトピー論シンポジウム2024  2024/06
• カンドルの付随群について  [Invited]

  秋田利之

  研究集会「カンドルと対称空間」  2024/01
• 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  2023/09
• カンドルのassociated groupについて  [Not invited]

  秋田利之, 長谷川蒼

  日本数学会秋季総合分科会  2022/09
• ブレイド群の中心拡大  [Invited]

  秋田利之

  代数的位相幾何学の軌跡と展望  2022/03
• The adjoint group of a Coxeter quandle  [Not invited]

  Toshiyuki Akita

  日本数学会2021年度年会  2021/03
• Artin群とCoxeterカンドルの随伴群のコホモロジー  [Invited]

  秋田利之

  森本雅治先生退職記念研究集会  2020/02
• On the cohomology of Coxeter groups and related groups  [Not invited]

  Toshiyuki Akita

  The Third Pan Pacific International Conference on Topology and Applications (PPICTA)  2019/11  成都(中国)
• Central extensions of Coxeter groups and Artin groups  [Invited]

  Toshiyuki Akita

  Branched Coverings, Degenerations, and Related Topics 2019  2019/03  広島大学(東広島キャンパス)大学院理学研究科
• Coxeter群とArtin群のコホモロジーについて  [Invited]

  秋田利之

  九州大学トポロジー金曜セミナー  2018/12
• Coxeterカンドルの随伴群  [Invited]

  秋田利之

  九州大学数理談話会  2018/12
• Coxeterカンドルとルート系の随伴群  [Invited]

  秋田利之

  2018年度ホモトピー論シンポジウム  2018/11
• Coxeterカンドルとルート系の随伴群  [Invited]

  秋田利之

  ホモトピー沖縄  2018/09
• Coxeter groups, Coxeter quandles, and Artin groups  [Invited]

  秋田利之

  Matroids, reflection groups, and free hyperplane arrangements  2018/06  RIMS
• Coxeter groups, Artin groups and Coxeter quandles  [Invited]

  Toshiyuki Akita

  研究集会「ストリングトポロジーとその周辺」  2017/12  四季の湯強羅静雲荘
• カンドルと対称群の中心拡大  [Invited]

  秋田利之

  北海道大学数学教室談話会  2017/10
• On the mod p cohomology of Coxeter groups and their alternating subgroups  [Not invited]

  Toshiyuki Akita

  ホモトピー論シンポジウム  2016/11  県立広島大学サテライトキャンパス
• Second mod 2 homology of Artin groups  [Invited]

  Toshiyuki Akita

  東大火曜トポロジーセミナー  2016/11
• Cohomology of Coxeter groups and related groups  [Invited]

  Toshiyuki Akita

  Perspectives on arrangements and configuration spaces  2016/09  Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore di Pisa
• Crossed modules and Artin groups  [Invited]

  Toshiyuki Akita

  京大代数トポロジーセミナー  2016/06

Association Memberships

• 日本数学会   

Research Projects

• 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

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2024/04 -2028/03 

  Author : 秋田 利之
• Cohomology of Coxeter groups, Artin groups, and Coxeter quandles

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2020/04 -2024/03 

  Author : 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。
• Topological studies on cohomology of Artin groups and related topics

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2017/04 -2020/03 

  Author : 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.
• Cohomology of mapping class groups, Coxeter groups and Artin groups

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2014/04 -2017/03 

  Author : 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.
• Constructions of cocycles of finite groups by characteristic classes

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2011 -2013 

  Author : 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.
• Coho mology of discrete groups and characteristic classes of flat bundles

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2008 -2011 

  Author : 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.
• Geometry of Groups and Moduli Spaces (2)

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2007 -2009 

  Author : 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.
• Topological Studies around Riemann Surfaces

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2006 -2009 

  Author : 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.
• Equivariant theory of Chern classes, Integrals based on Euler characteristics and related topics

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2005 -2008 

  Author : 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.
• Studies on the Cohomology of Mapping Class Groups, Coxeter Groups and Artin Groups

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2005 -2007 

  Author : 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.
• Applied singularity theory to exterior differential systems

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2002 -2005 

  Author : 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.
• 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

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2002 -2004 

  Author : 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。
• Complex analytic approach towards topology studies on the mapping class ganups for surfaces

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2002 -2004 

  Author : 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.
• Mapping Class Group of Surfaces and Geometry of Moduli Spaces

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2001 -2003 

  Author : 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.
• 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

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2000 -2001 

  Author : 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。
• A topological study of the moduli space of Riemann surface

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1999 -2001 

  Author : 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.
• 反復的曲面束の位相幾何学的研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1997 -1998 

  Author : 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。
• On the construction and classification of the finite geometry

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1997 -1998 

  Author : 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).
• Homotoy Theory of Classifying Spaces

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1997 -1998 

  Author : 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.
• 正曲率及びワイル共形曲率と多様体の空間

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1996 -1996 

  Author : 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。