How did Poincare discover the fundamental group? What was the first instance that led Poincaré to discover this amazing theory?


The concept of fundamental group was introduced in Poincaré's seminal paper Analysis situs, so that is a natural place to look for some motivation for this concept. In Stillwell's nice translation of Poincaré's Analysis situs and related papers he explains the motivation in one of the introductory paragraphs, written by himself:

Poincaré set the agenda for his 1895 Analysis situs paper with a short announcement [...] In it he 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.

Fortunately, Stillwell also included a translation of this anouncement of Analysis situs. In it, after Poincaré introduced the concept of Betti numbers, we find the following paragraphs (with footnotes from Stillwell):

One may ask whether the Betti numbers suffice to determine a closed surface from the viewpoint of Analysis situs. That is, given two surfaces with the same Betti numbers, we ask whether it is possible to pass from one to the other by a continuous deformation. This is true in the space of three dimensions, and we may be inclined to believe that it is again true in any space. The contrary is true.

In order to explain, I want to approach the question from a new viewpoint. Let $$x_1, x_2,\dots,x_{n+1}$$ be the coordinates of a point on the surface. These $n+1$ quantities are connected by the equation of the surface. Now let $$F_1, F_2,\dots,F_p$$ be any $p$ functions of the $n + 1$ coordinates $x$ (which I always suppose to be connected by the equation of the surface, and which I suppose to take only real values).

I do not assume that the functions $F$ are uniform, but I suppose that if the point $(x_1 , x_2 , \dots , x_{n+1})$ describes an infinitely small contour on the surface then each of the functions $F$ returns to its initial value. This being so, we suppose that our point now describes a finite closed contour on the surface. It may then happen that the $p$ functions do not return to their initial values, but instead become $$F_1',F_2',\dots, F_p'$$

In other words, they undergo the substitution $$(F_1,F_2,\dots,F_p;F_1',F_2',\dots, F_;').$$ All the substitutions corresponding to the different closed contours that we can trace on the surface form a group which is discontinuous (at least as far as its form is concerned).

This group evidently depends on the choice of functions $F$. We suppose first that these functions are the most one can imagine, other than being subject to the condition imposed above, and let $G$ be the corresponding group. If $G'$ is the group corresponding to another choice of functions, then $G'$ will be isomorphic to $G$—holoedrically in general but meriedrically in special cases$^1$.

The group $G$ can then serve to define the form of the surface and it is called the group of the surface$^2$. It is clear that if two surfaces can each be transformed to the other by a continuous transformation, then their groups are isomorphic. The converse, though less evident, is again true for closed surfaces, so that what defines a closed surface, from the viewpoint of Analysis situs, is its group$^3$.

$^1$ The terminology "holoedric isomorphism" and "meriedric isomorphism" correspond to the modern concepts of isomorphism and homomorphism, respectively.

$^2$ This would later be called the fundamental group in §12 of Analysis situs itself.

$^3$ This is true for surfaces in the traditional, two-dimensional, sense, but not for manifolds of three dimensions. Near the end of §14 of Analysis situs, Poincaré raised the question whether two manifolds with the same group are necessarily homeomorphic.

In conclusion, it appears that the concept appeared quite naturally in Poincaré's attempts to classify surfaces/manifolds up to homeomorphism.

  • $\begingroup$ I think in the text the terms holoedrically and meriedically have to be transposed. (This typo seems to be there in Stillwell's translation as well) $\endgroup$ – shvjds Mar 22 '16 at 16:55

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