Studies on the rational homology of free loop spaces

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13403
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Nippon Institute of Technology
• Principal Investigator: Naito Takahito 日本工業大学, 共通教育学群, 講師 (20724511)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 自由ループ空間 / ストリングトポロジー / 有理ホモトピー論

Outline of Final Research Achievements

The free loop space is a topological space consisting of all continuous map from the circle to a space. The aim of this research is to investigate rational homology of free loop spaces and obtain geometric information of a given manifold from algebraic structures in string topology. As consequence, we have the following research results. (1) we observe a generator set of the loop homology algebras and, moreover, give a new computational method of the rational string bracket. (2) we introduce Cartan calculi on the free loop spaces and clarify relations between the calculus and the algebraic structures in string topology. (3) we study the string coproduct on the reduced loop homology by rational homotopy theory. Especially, we investigate a behavior of the coproduct of pure manifolds with respect to the degree of the Hodge decomposition of the loop homology.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

自由ループ空間のホモロジーや、その上のストリングトポロジー代数構造を計算する事は一般的に困難である。本研究成果では、まずBV完全という新たな位相空間のクラスを導入し、同変ループホモロジー上のストリング括弧積の新たな計算方法を与えた。また自由ループ空間に関するCartan calculusとストリングトポロジー代数構造を関連付け、自由ループ空間のホモロジー研究を推進したと同時に、新たな幾何学的研究アプローチを発見することも出来た。