Some Studies on the Motivations of Poincaré's Topological Thought

Jules Henri Poincaré (1854-1912) is regarded as “the last universalist”. It is well known that one of his most significant contributions to mathematics is the creation of algebraic topology. Based on primary and secondary literature, combined with some relevant works written by famous mathematicians and historians of mathematics, this paper not only reveals and corrects some defects on the motivations of Poincaré's topology in previous research, but also expounds the motivations of Poincaré topology research more comprehensively and systematically. It further points out that the Poincaré Conjecture is also a potential motivation for his topology research. At the same time, in this paper, we analyze Poincaré's application of relevant topological results and the status of Poincaré Conjecture in his entire topological work, showing that it reflects the universality of Poincaré's mathematical research field.