同伦论的分析学溯源

The Analytical Origin of Homotopy Theory

同伦论是代数拓扑学的重要研究内容,其思想起源可以追溯到分析学中的许多问题。柯西在研究复函数的积分值与积分路径的关系时提出了柯西积分定理,黎曼为处理多值函数而引入的黎曼面,皮瑟在处理代数函数时引入的连续变换思想以及若尔当有关闭曲线的研究等,都为庞加莱定义基本群产生了非常重要的影响。通过研究同伦论的分析学溯源,可以使我们更清楚地认识到同伦论的历史演变过程,同时为理解近现代几何学的发展状况提供一个窗口。

The homotopy theory is an important research content in algebraic topology,and its ideological origin can be traced back to many problems from analytics. The Cauchy integral theorem proposed by Cauchy in studying the relationship between the integral values of complex functions and the integral paths,the Riemann surface introduced by Riemann to deal with multivalued functions,and the continuous transformation thought introduced by Puiseux to deal with algebraic functions and the studies of the closing curve of Jordan,all have had a very important influence on the definition of the fundamental group of Poincaré. By studying the analytical origin of homotopy theory,we can better understand the historical evolution of homotopy and provide a window for understanding the development of modern geometry.