The set of paths in a space and its algebraic structure. A historical account. (English. French summary) Zbl 1382.01006
This paper investigates the significance of the historical concept of ‘structuralism’ in the history of modern mathematics by considering how Poincaré’s notion of fundamental group (of homotopy classes of closed paths in a topological space) was developed by subsequent generations. The paper begins with a survey of Poincaré’s various approaches to fundamental groups, before moving to the ways in which the concept was handled after Poincaré, in for example the work of Weyl, Schreier, and Reidemeister. The extension of the notion of fundamental group to that of a fundamental groupoid (of general paths) comes next, followed by the study of finer notions of equivalence than homotopy. It is noted that each of the latter developments has been used only marginally by mathematicians, compared to the original fundamental group, and reasons are given for this.
Reviewer: Christopher Hollings (Oxford)
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