In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré’s works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not been addressed either. For example, one question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré’s works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré’s stated motivations for topology and the Poincaré conjecture.

在过去的十年中，庞加莱猜想可能是庞加莱对数学界的所有贡献中最著名的陈述。有许多论文和书籍描述了佩雷尔曼的各种尝试和最终作品导致了该猜想的积极解决，但是庞加莱作品导致该猜想的演变尚未得到仔细讨论或描述，并且关于庞加莱作品的演变还没有被仔细讨论或描述，并且关于这个问题也没有得到解决。例如，一个问题是它如何融入庞加莱拓扑学的整体工作，以及他提出的其他相关问题是什么。由于庞加莱并没有将庞加莱猜想作为一个猜想来表述，而是将其作为一个问题提出来，一个自然的问题是他为什么要这样做。为了解决这些问题，在本文中，我们在通过数值和代数不变量对流形进行分类的框架中研究了庞加莱的拓扑学著作。因此，我们还提供了庞加莱猜想的完整历史，它比通常描述和接受的更丰富，因此可以更好地理解庞加莱拓扑学的整体著作。此外，这一分析还澄清了庞加莱所阐述的拓扑动机与庞加莱猜想之间关系的一个令人费解的问题。