庞加莱引入立方体流形动因探析  

An Analysis of the Motivation Behind Poincaré's Introduction of Cube Manifolds

作　　者：冯慧敏 刘娜娜 王昌[1] FENG Huimin;LIU Nana;WANG Chang(Institute for Advanced Study in History of Science,Northwest University,Xi'an 710127,China)

机构地区：[1]西北大学科学史高等研究院,西安710127

出　　处：《中国科技史杂志》2023年第2期228-239,316,共13页The Chinese Journal for the History of Science and Technology

基　　金：国家自然科学基金资助项目“同伦论的历史研究”(项目编号:11501444)。

摘　　要：庞加莱在关于位置分析的一系列论文(1892—1904)中提出贝蒂数、挠系数、基本群、欧拉-庞加莱示性数等不变量,试图对所有维数的流形进行分类。在庞加莱同调理论发展的过程中,他构造了许多闭的3维流形,这些流形发挥了重要的作用。其中一类重要的3维流形,称为“立方体流形”,它们是通过等同立方体的表面来构造的。这个想法可以追溯到庞加莱在1880年代初期对自守函数理论的研究。本文将阐明庞加莱为什么引入立方体流形,以及立方体流形是如何影响庞加莱同调理论发展的。A series of papers(1892—1904)written by Poincaréon Analysis Situs in which he proposed invariants such as Betti numbers,torsion coefficients,fundamental group and Euler-Poincarécharacteristic attempted to classify manifolds of all dimensions.In the course of the development of the homology theory of Poincaré,he constructed many closed 3-dimensional manifolds,which played an important role.One important class of 3-dimensional manifolds,called“cube manifolds”,is constructed by identical surfaces of the cube.This idea can be traced back to Poincaré's work on the theory of automorphic functions in the early 1880s.In this paper,we will clarify why Poincaréintroduced cube manifolds and how cubic manifolds influenced the development of Poincaré's work on homology theory.

关 键 词：贝蒂数 基本群 立方体流形 自守函数 

分 类 号：N09[自然科学总论—科学技术哲学] O1-0[理学—数学]