Fundamental group

Japanese: 基本群 - きほんぐん（英語表記）fundamental group

The fundamental group, along with the homology group, is a concept introduced by H. Poincaré to study the topological properties of figures algebraically, and is also called the one-dimensional homotopy group. It is defined for a figure X and a point x , and is represented by π ₁ ( X , x ). Consider all closed curves on X that go from x to x and return to x , and group all closed curves that can be transferred to each other by a continuous transformation without moving x into the same class. This class is an element of π ₁ ( X , x ), and the product of the class of closed curve f and the class of closed curve g is defined as the class of closed curves connecting f and g (Figure).

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

基本群は図形の位相的性質を代数的に研究するために，ホモロジー群とともに，H.ポアンカレによって導入された概念で，一次元ホモトピー群とも呼ばれる。これは図形Xとその1点xに対して定義され，π₁(X,x)で表される。xを出てxにもどるX上の閉曲線全体を考え，そしてxを動かさない連続的変形により互いにうつりうる閉曲線は同じ類にまとめる。この類がπ₁(X,x)の元で，閉曲線fの類と閉曲線gの類の積はfとgをつないだ閉曲線の類として定義される(図)。

出典　株式会社平凡社世界大百科事典第２版について　情報

<<: Gibbons, Orlando

>>: Basic salary - Kihonkyu