Research on smooth structures on 4-manifolds using handle decompositions

• Project/Area Number: 25800048
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Hiroshima University
• Principal Investigator: YASUI Kouichi 広島大学, 理学(系)研究科(研究院), 助教 (70547009)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: トポロジー / 4次元多様体 / 微分構造 / ハンドル分解 / Stein構造 / 接触構造 / コルク / 結び目 / ハンドル体 / Stein filling / レフシェッツファイブレーション / Lefschetz fibration

Outline of Final Research Achievements

We gave an algorithm which produces infinitely many pairwise exotic (i.e. homeomorphic but non-diffeomorphic) Stein fillings of the same contact 3-manifolds. We also gave a method for producing pairs of framed knots which represent exotic pairs of 4-manifolds. As an application, we disproved the Akbulut-Kirby conjecture on knot concordance. We also obtain results on contact 5-manifolds and maximal Thurston-Bennequin numbers of knots.

Research Products

• [Int'l Joint Research] ミシガン州立大学(米国)
• [Journal Article] Maximal Thurston-Bennequin number and reducible Legendrian surgery (2016)
  • Author(s): Kouichi Yasui
  • Journal Title: Compositio Mathematica Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Nonexistence of Stein structures on 4-manifolds and maximal Thurston-Bennequin numbers (2016)
  • Author(s): Kouichi Yasui
  • Journal Title: Journal of Symplectic Geometry Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Contact 5-manifolds admitting open books with exotic pages (2016)
  • Author(s): Selman Akbulut and Kouichi Yasui
  • Journal Title: Mathematical Research Letters Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Infinitely many small exotic Stein fillings (2014)
  • Author(s): Selman Akbulut and Kouichi Yasui
  • Journal Title: Journal of Symplectic Geometry Volume: 4 Issue: 4 Pages: 673-684
  • DOI: 10.4310/jsg.2014.v12.n4.a2
  • Peer Reviewed
• [Journal Article] Gluck twisting 4-manifolds with odd intersection form (2013)
  • Author(s): Selman Akbulut and Kouichi Yasui
  • Journal Title: Mathematical Research Letters Volume: 02 Issue: 2 Pages: 385-389
  • DOI: 10.4310/mrl.2013.v20.n2.a13
  • Peer Reviewed
• [Presentation] Stein ４次元多様体の微分構造と５次元多様体の接触構造 (2016)
  • Author(s): 安井弘一
  • Organizer: 微分トポロジー１６
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-22
  • Invited
• [Presentation] Maximal Thurston-Bennequin number and reducible Legendrian surgery (2016)
  • Author(s): 安井弘一
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-18
• [Presentation] Nonexistence of Stein structures on 4-manifolds and maximal Thurston-Bennequin numbers (2016)
  • Author(s): 安井弘一
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-18
• [Presentation] Corks, exotic 4-manifolds and knot concordance (2016)
  • Author(s): 安井弘一
  • Organizer: Branched Coverings, Degenerations, and Related Topics 2016
  • Place of Presentation: 広島大学
  • Year and Date: 2016-03-10
  • Int'l Joint Research / Invited
• [Presentation] Contact 5-manifolds admitting open books with exotic Stein page (2016)
  • Author(s): 安井弘一
  • Organizer: 接触構造，特異点，微分方程式及びその周辺
  • Place of Presentation: 金沢大学
  • Year and Date: 2016-01-23
• [Presentation] Corks, exotic 4-manifolds and knot concordance (2015)
  • Author(s): 安井弘一
  • Organizer: 研究集会「４次元トポロジー」
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2015-11-21
• [Presentation] Contact 5-manifolds admitting open books with exotic Stein pages (2015)
  • Author(s): Selman Akbulut, 安井弘一
  • Organizer: 日本数学会2015年度秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-15
• [Presentation] Corks, exotic 4-manifolds and knot concordance (2015)
  • Author(s): 安井弘一
  • Organizer: 日本数学会2015年度秋季総合分科会
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-15
• [Presentation] Corks, exotic 4-manifolds and knot concordance (2015)
  • Author(s): 安井弘一
  • Organizer: Friday Seminar on Knot Theory
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2015-06-26
  • Invited
• [Presentation] Exotic 4-manifolds and non-concordant knots with the same 0-surgery (2015)
  • Author(s): 安井弘一
  • Organizer: 広島大学トポロジー・幾何セミナー
  • Place of Presentation: 広島大学
  • Year and Date: 2015-04-21
• [Presentation] Partial twists and exotic Stein fillings (2015)
  • Author(s): 安井弘一
  • Organizer: 微分トポロジー１５～微分トポロジーの過去・現在・未来～
  • Place of Presentation: 京都大学
  • Year and Date: 2015-03-25
  • Invited
• [Presentation] Partial twists and exotic Stein fillings (2014)
  • Author(s): Kouichi Yasui
  • Organizer: Topology Seminar
  • Place of Presentation: Kansas State University （アメリカ）
  • Year and Date: 2014-11-14
  • Invited
• [Presentation] Corks and exotic smooth structures on 4-manifolds (2014)
  • Author(s): Kouichi Yasui
  • Organizer: Colloquium
  • Place of Presentation: Kansas State University （アメリカ）
  • Year and Date: 2014-11-13
  • Invited
• [Presentation] Partial twists and exotic Stein fillings (2014)
  • Author(s): Kouichi Yasui
  • Organizer: Topology Seminar
  • Place of Presentation: Michigan State University （アメリカ）
  • Year and Date: 2014-10-06
• [Presentation] Partial twists and exotic Stein fillings (2014)
  • Author(s): Kouichi Yasui
  • Organizer: Differential Geometry and Symplectic Topology Seminar
  • Place of Presentation: University of Minnesota （アメリカ）
  • Year and Date: 2014-09-25
  • Invited
• [Presentation] Partial twists and exotic Stein fillings (2014)
  • Author(s): Kouichi Yasui
  • Organizer: Workshop on Topology and Invariants of 4-Manifolds
  • Place of Presentation: Simons Center for Geometry and Physics, Stony Brook University （アメリカ）
  • Year and Date: 2014-08-26
  • Invited
• [Presentation] Partial twists and exotic Stein fillings (2014)
  • Author(s): 安井弘一
  • Organizer: 東工大トポロジーセミナー
  • Place of Presentation: 東京工業大学
  • Year and Date: 2014-07-09
• [Presentation] Partial twists and exotic Stein fillings (2014)
  • Author(s): 安井弘一
  • Organizer: 広島大学トポロジー・幾何セミナー
  • Place of Presentation: 広島大学
  • Year and Date: 2014-06-24
• [Presentation] Partial twists and exotic Stein fillings (2014)
  • Author(s): 安井弘一
  • Organizer: 幾何学コロキウム
  • Place of Presentation: 北海道大学
  • Year and Date: 2014-06-13
• [Presentation] Partial twists and exotic Stein fillings (2014)
  • Author(s): 安井弘一
  • Organizer: 日本数学会2014年度年会
  • Place of Presentation: 学習院大学
• [Presentation] Stein 構造の4 次元多様体への応用についてII (2014)
  • Author(s): 安井弘一
  • Organizer: 接触構造、特異点、微分方程式及びその周辺
  • Place of Presentation: 高知市文化プラザかるぽーと
• [Presentation] Stein 構造の4 次元多様体への応用についてI (2014)
  • Author(s): 安井弘一
  • Organizer: 接触構造、特異点、微分方程式及びその周辺
  • Place of Presentation: 高知市文化プラザかるぽーと