Study of higher homotopy commutativity of loop spaces by constructing polytopes

2014 Fiscal Year Final Research Report

• Project/Area Number: 24740053
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: 防衛大学校(総合教育学群、人文社会科学群、応用科学群、電気情報学群及びシステム工学群)
• Principal Investigator: KAWAMOTO Yusuke 防衛大学校(総合教育学群、人文社会科学群、応用科学群、電気情報学群及びシステム工, 総合教育学群, 准教授 (10531759)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: ループ空間 / ホップ空間 / 高位ホモトピー可換性 / 巡回多面体 / 置換結合多面体

Outline of Final Research Achievements

The concept of higher homotopy commutativity was first studied by Sugawara in the case of topological monoids. In the present study, we constructed a new higher homotopy commutativity for the (loop) multiplications of loop spaces (higher homotopy associative Hopf spaces) using cyclohedra constructed by Bott and Taubes. We also gave combinatorial decompositions of permuto-associahedra constructed by Kapranov into unions of product spaces of cyclohedra. Then the cyclohedron can be regarded as a subspace of the permuto-associahedron. From the decompositions, we also studied relations between the new concept of higher homotopy commutativity and another one represented by the permuto-associahedra.

Free Research Field

代数的位相幾何学