Paris: Gauthier-Villars et Fils, 1892.
First edition of Poincaré’s first paper on topology, which gave birth to the subject in its modern sense. “One of the most unexpected developments in twentieth-century mathematics has been the meteoric rise of the subject known as topology. Topology is sometimes described as ‘rubber-sheet geometry’, a whimsical and somewhat misleading description which nevertheless succeeds in capturing the flavour of the subject. Topology is the study of those properties of geometrical objects which remain unchanged under continuous transformations of the object” (Stewart, p. 144). “The publication (1892) of Poincaré’s first note on topology … marks the beginning of the subject of algebraic, or ‘combinatorial,’ topology. There was earlier scattered work by Euler, Listing (who coined the work ‘topology’), Möbius and his band, Riemann, Klein, and Betti. Indeed, even as early as 1679, Leibniz indicated the desirability of creating a geometry of the topological type. The establishment of topology (or ‘analysis situs’ as it was often called at the time) as a coherent theory, however, belongs to Poincaré” (Bredon, p. v). “The development of mathematics in the nineteenth century began under the shadow of a giant, Carl Friedrich Gauss; it ended with the domination by a genius of similar magnitude, Henri Poincaré” (DSB XI: 52).
“It was Poincaré who both made the pioneering steps in topology and forecast that topology would be an important ingredient in 20th-century mathematics. Incidentally, Hilbert, who made his famous list of problems, did not. Topology hardly figured in his list of problems. But for Poincaré it was quite clear that it would be an important factor” (Atiyah, pp. 654-5). Indeed, Poincaré writes in the present paper: “As for me, all of the diverse paths on which I was successively engaged led me to analysis situs. I had need of the ideas of this science to pursue my studies on curves defined by differential equations ... and in particular for those of the three-body problem. I had need of it for the study on multivalued functions of two variables. I had need of it for the study of periods of multiple integrals and for the application of this study to the development of the perturbation function. Finally I glimpsed in analysis situs a means of attacking an important problem in the theory of groups, the search for discrete or finite groups contained in a given continuous group” (pp. 633-4).
“Without much exaggeration, it can be said that only one important topological concept came to light before Poincaré. This was the Euler characteristic of surfaces, whose name stems from the paper of Euler (1752) on what we now call the Euler polyhedron formula … Between the 1820s and 1880s, several different lines of research were found to converge to the Euler characteristic … All of these ideas admit generalizations to higher dimensions, but the only substantial step towards topology in arbitrary dimensions before Poincaré was that of Betti (1871). Betti was inspired by Riemann’s concept of connectivity of surfaces to define connectivity numbers, now known as Betti numbers P1, P2, . . . , in all dimensions. The connectivity number of a surface S may be defined as the maximum number of disjoint closed curves that can be drawn on S without separating it.
“In the introduction to his first major topology paper, the Analysis situs, Poincaré announced his goal of creating an n-dimensional geometry. As he memorably put it: “geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset.” Because “positions must not be upset,” Poincaré sought what Leibniz called Analysis situs, a geometry of position, or what we now call topology. He cited as precedents the work of Riemann and Betti, and his own experience with differential equations, celestial mechanics, and discontinuous groups …
“In [the present paper, Poincaré] raises the question whether the Betti numbers suffice to determine the topological type of a manifold, and introduces the fundamental group to further illuminate this question. He gives a family of three-dimensional manifolds … and shows that certain of these manifolds have the same Betti numbers but different fundamental groups. It follows, assuming that the fundamental group is a topological invariant, that the Betti numbers do not suffice to distinguish three-dimensional manifolds” (Stillwell, pp. 1-6).
In a long series of subsequent papers on Analysis situs, Poincaré developed the ideas outlined in this first announcement in several directions. He attempted to provide a new foundation for the Betti numbers in a rudimentary ‘homology theory’, which introduces the idea of computing with topological objects. Using his homology theory, he discovered a ‘duality theorem’ for the Betti numbers of an n-dimensional manifold. He generalized the Euler polyhedron formula to arbitrary dimensions and situated it in his homology theory. Recognizing that the fundamental group first becomes important for three-dimensional manifolds, Poincaré asked whether it suffices to distinguish between them. Poincaré was unable to answer this last question. A special case is the so-called Poincaré conjecture, which asks whether a (closed) three-dimensional manifold with trivial fundamental group is topologically equivalent to the three-dimensional sphere. This extremely difficult question was answered in the affirmative by Grigori Perelman in 2002.
Michael F. Atiyah, ‘Mathematics in the 20th century,’ American Mathematical Monthly 108 (2001), 654-666; Enrico Betti, ‘Sopra gli spazi di un numero qualunque di dimensioni,’ Annali di Matematica pura ed applicate 4 (1871), 140–158; Glen E. Bredon, Topology and Geometry, 1993; Leonhard Euler, ‘Elementa doctrinae solidorum,’ Novi Comm. Acad. Sci. Petrop. 4 (1752), 109–140; Bernhard Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, 1851; Dirk Siersma, ‘Poincaré and Analysis Situs, the beginning of algebraic topology,’ Nieuw Archief voor Wiskunde 5/13 (2012), 196-200; Ian Stewart, Concepts of Modern Mathematics, 1995; John Stillwell, Henri Poincare. Papers on Topology, 2010.
Pp. 633-636 in Comptes Rendus hebdomadaires de l'Académie des sciences de Paris, Tome CXV, No. 18, 31 Octobre 1892. Paris: Gauthier-Villars et Fils, 1892. 4to (287 x 230 mm), pp. 633-696. Original pink string-bound printed wrappers. Offered here in the rare weekly issue (the anual volumes are much commoner). Spine strip with some fraying and tears.