Topology of the embedding spaces via configuration space integrals, operads and the calculus of functors

• Project/Area Number: 23840015
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Shinshu University
• Principal Investigator: SAKAI Keiichi 信州大学, 理学部, 助教 (20466824)
• Project Period (FY): 2011 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2011: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 埋め込みの空間 / グラフ / オペラッド / 位相的Stiefel多様体 / ストリング・トポロジー / BV代数 / Hochschildホモロジー / 配置空間 / 配置空間積分 / グラフ複体 / 関手の微積分

Research Abstract

I continued the study of the spaces of embedding between spheres. I showed that the space fE called “the space of embeddings modulo immersions” is an iterated loop space of the topological Stiefel manifold. As applications I obtained;
(1) A BV-algebra structure on the homology of fE
(2) An alternative proof in terms of topological Stiefel manifolds of the fact that the “spinning” operation gives rise to an isomorphism of the lower dimensional homotopy groups of the embedding spaces.

Research Products

• [Journal Article] 1-loop graphs and configuration space integral for embedded spaces (2012)
  • Author(s): Keiichi Sakai, Tadayuki Watanabe
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society Volume: 152 Issue: 3 Pages: 497-533
  • DOI: 10.1017/s0305004111000429
  • NAID: 120005228644
  • Peer Reviewed
• [Presentation] 結び目の種数を減らすある変形について (2013)
  • Author(s): 境圭一
  • Organizer: Spring Workshop 2013 on Low-Dimensional Topology and its Ramifications
  • Place of Presentation: 名城大学名駅サテライト
  • Year and Date: 2013-03-02
• [Presentation] 枠つきlong embeddingの空間のdeloopingについて (2013)
  • Author(s): 境圭一
  • Organizer: 幾何学セミナー
  • Place of Presentation: 首都大学東京
  • Year and Date: 2013-01-11
• [Presentation] BV structures on the homology of the framed long embedding spaces (2012)
  • Author(s): 境圭一
  • Organizer: 空間の代数的・幾何的モデルとその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-09-11
• [Presentation] Embedding spaces and string topology (2012)
  • Author(s): 境圭一
  • Organizer: 講演会
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2012-05-08
• [Presentation] BV-structures on the homology of the framed long knot space (2011)
  • Author(s): Keiichi Sakai
  • Organizer: 4th East Asian Conference on Algebraic Topology
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2011-12-09
• [Presentation] BV-structures on the homology of the framed long knot space (2011)
  • Author(s): Keiichi Sakai
  • Organizer: The Fourth East Asian Conference on Algebraic Topology
  • Place of Presentation: 東京大学大学院数理科学研究科(招待講演)
  • Year and Date: 2011-12-09
• [Presentation] Embedding spaces and string topology
  • Author(s): 境 圭一
  • Organizer: 講演会
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Invited
• [Presentation] 枠つきlong embeddingの空間のdeloopingについて
  • Author(s): 境 圭一
  • Organizer: 幾何学セミナー
  • Place of Presentation: 首都大学東京
  • Invited
• [Remarks]
  • URL: http://math.shinshu-u.ac.jp/~ksakai/index_j.html
• [Remarks] Website of Keiichi Sakai
  • URL: http://math.shinshu-u.ac.jp/~ksakai/index_j.html
• [Remarks]
  • URL: http://math.shinshu-u.ac.jp/~ksakai/index_j.html