What specific problems motivated Poincaré's work on topology?

Question

The MacTutor biography on Poincaré says:

Poincaré's Analysis situs, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology. For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work.

The fundamental group is credited to Poincaré, plus much of homology theory.

What specific problems stimulated his topological work? Was it an outgrowth of his work on celestial mechanics, or automorphic functions, or something else? The McTutor article mentions that he introduced the fundamental group to help classify two-dimensional surfaces, but why was he trying to do that?

Answer 1

A germane and interesting article on the topic:

• Ji, L., & Wang, C. (2020). Poincaré’s stated motivations for topology. Archive for History of Exact Sciences, 74, 381-400.

I won't recap its content here, but it is a detailed answer to the question.

While Picard's influence is mentioned I would like to point out that this goes back to Poincare's very first paper on analysis situs of (1892) (using Stillwell's translation) where he writes the following:

Persons who recoil from geometry of more than three dimensions may believe this result to be useless and view it as a futile game, if they have not been informed of their error by the use made of Betti numbers by our colleague M. Picard in pure analysis and ordinary geometry.

While Poincare does not provide a citation there is little doubt that he is refering to:

• Picard, E. (1889). Memoire sur la theorie des fonctions algebriques de deux variables. J. de Math. 5, 135–319. Discussion of analysis situs starts at page 179.

He closes this very paper as follows:

This remark throws some light on the theory of ordinary algebraic surfaces and makes less strange the discovery of M. Picard, according to which the surfaces have no one-dimensional cycle if they are the most general of their degree.

This suggests that building on Picard's work of using Betti numbers and notions of cycles is an articulated motivation of Poincare as early as 1892.

Furthermore Picard and Simart (1897) were quick to adopt Poincare's work on analysis situs:

• Picard, E., Simart, G. (1897) Theorie des fonctions algebriques de deux variables independantes, vol. I. Gauthier-Villars et Fils. See chapter 2.

Another aspect that is well researched and of interest is Poincare's early use of work of Riemann and Betti. Consider the fascinating:

• Mawhin, J. (2000). Poincaré's early use of Analysis situs in nonlinear differential equation: variations around the theme of Kronecker's integral. Philosophia Scientiae, 4(1), 103-143.

My favorite example is Poincare (1886)

Poincaré, H. (1886). Sur les courbes définies par les équations différentielles (quatrième partie). Journal de mathématiques pures et appliquées, 2, 151-217.

On p. 186 he cites Riemann and "Brioschi" as a misnaming of Betti. However this fun observation aside both with Poincare in this case (1886) and Picard of 1889, Riemann's and Betti's ideas were being considered in differential and algebraic geometric settings. By 1895 Poincare had cracked the nut and seen the pattern that underlies these notions of Riemann/Betti.

In hindsight forms of algebraic/differential topology might even seem as somewhat inevitable. If you study surfaces (as Picard did) you will have cases where (co)homology enters and theorems are discovered that some curves on the surface must have certain global properties (the topological ones!). Mawhin's article goes through a range of interesting post-hoc reinterpretations of results in this spirit. Regardless it is remarkable how Poincare stamped out a whole theory and defined many fundamental notions out of what previously appears to be rather very meager starting points. Some of what we today might call "application areas" were the original problem spaces under investigation. For example:

• Dieudonné, J. A. (1989). A history of algebraic and differential topology, 1900-1960 (pp. 598-600). Boston: Birkhäuser.

In the chapter "Applications of Homology to Geometry and Analysis" he writes:

As early as 1888 Picard was using Betti's "orders of connectivity" and "deforming two-dimensional cycles" on an algebraic surface considered as a four-dimensional real "variety." His most interesting results can be deduced from a method he invented, which will later be called the study of a "pencil" of algebraic curves on an irreducible surface S.

In other words another possible theory for "motivation" is actually a kind of emergence from specifics in the problems studied, which in turn explains the kind of applications of homology before homology as given by Dieudonne.

Answer 2

One motivation, perhaps the principal one, was his work in ordinary differential equations. (And celestial mechanics, as an application of ordinary differential equations). He introduced what is called "qualitative methods" which are based to a large extend on topology. For example, the existence of periodic orbits.

In the introduction to his paper Analysis Situs (1895) Poincare mentions the following motivations: classification of algebraic surfaces, qualitative theory of differential equations, and applications to what is known now as Lie groups.

Answer 3

Specifically, possible motivation would have been provided by the Poincare recurrence theorem and/or the Poincare-Hopf theorem. The validity of these results depend on global hypotheses on the manifold.