Inifinitary generated objects(Algebraic topology of wild spaces)

2022 Fiscal Year Final Research Report

• Project/Area Number: 15K04882
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Waseda University
• Principal Investigator: Eda Katsuya 早稲田大学, 理工学術院, 名誉教授 (90015826)
• Project Period (FY): 2015-04-01 – 2023-03-31
• Keywords: Fundamental groups / wild algebraic topology / one dimensional space / infinitary words / singular homology

Outline of Final Research Achievements

1.Cortorsionfree abelian groups are defined by the countable direct product of integer group. As non-commutative version of this we define groups using the Hawaiian earring group. Then, for abelian groups this new notion coincides with the cotorsionfreeness.
2.If there exists an overlay over a locally compact group, then the total space becomes a topological group and the overlay map becomes a topological homomorphism.
3.Suppose that the wild parts of one-dimensional Peano continua are non-empty and 0-dimensional. If the wild parts of the spaces are homeomorphic, then the spaces are homotopy equivalent. Conversely, if the spaces are homotopy equivalent, their wild parts are homeomorphic. Any one-dimensional Peano continuum X there exists a locally finite connected graph such that the space with its ends realizes X. Conversely the space of locally finite connected graph is a one-dimensional Peano continuum with 0-dimensional wild part.

Free Research Field

Geometric Topology

Academic Significance and Societal Importance of the Research Achievements

代数的トポロジーの対象は従来、局所的によい空間に関するものであった、そのため非可算濃度をもつ群あるいはその性質が問題となることはなかった。研究代表者は1990年ころから野生的空間の基本群、特異ホモロジーの研究を始めた。今回の7年に渡る研究は、その大きな区切りであり、今後の Wild Algebraic Topology といわれる分野の確立である。これは、Infinite Abelian group でなされた非可算群の理論の新しい応用であり、またこれまでほぼなされていなかった非可換非可算群の研究とも考えられる。J. Brazas のホームページ Wild Topology に詳しい。