Poincaré declared his motivation for establishing a new branch of mathematics devoted to topology, or ‘the study of geometric objects undergoing deformation’, “rubber-sheet geometry”, in Analysis Situs with the statement (1895):

“On a bien souvent répété que la Géométrie est l’art de bien raisonner sur des figures mal faites; encoreces figures, pour ne pas nous tromper, doivent-elles satisfaire à certaines conditions; les proportions peuvant être grossièrement altérées, mais les positions relatives des diverses parties ne doivent pas être bouleversées.”

Namely, that there was a need for mathematicians to be able to determine with certainty that our encoreces figures ‘badly drawn figures’ must satisfy certain conditions such that even though their ‘proportions may be grossly altered’, that the ‘relative positions of the different parts must not be upset’. The statement aligns closely with the modern day definition of topology:

Topology is the study of the properties of a geometric object that is preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

On Analysis Situs (1892)

In his first paper (letter, really) on topology, Poincaré sets out to motivate the first real primer on topology, Analysis Situs (1895). He does so by referring to the Betti numbers introduced about 20 years earlier. His argument, or question to the reader, is whether Betti numbers actually suffice to determine the topological classification of a manifold. To show why they may not, he introduced the concept of a fundamental group π₁. Informally, a fundamental group may be thought of in the following way:

Start with a space (e.g. a surface), and some point in it, and all the loops both starting and ending at this point — paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined together in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space.

He next describes a family of three-dimensional manifolds and shows that certain of these manifolds have the same Betti numbers, yet belong to different fundamental groups (Stillwell 2010, p. 6). From this, he argues that if the fundamental group is a topological invariant (a property preserved while undergoing homeomorphisms), that Betti numbers alone cannot distinguish three-dimensional manifolds from one another.

Analysis Situs (1895)

The later Poincaré conjecture (1904) in fact did not exist in 1895, as according to Stillwell (2010), Poincaré at this point likely thought it obvious that all simply-connected n-dimensional closed manifolds would be homeomorphic to the n-sphere, i.e. that all such manifolds would preserve their topological properties if deformed to the shape of a sphere in n-dimensions. After all, the same result had been known to be true for 1- and 2-dimensional manifolds since the time of Riemann.

The Analysis Situs sets out, rather, to revise and supplement Betti numbers in search of a more solid foundation, given his own argumentation from three years prior. The paper works towards this goal by several paths. He begins by introducing, as often is the case in research, a justification for why the work is valuable, stating that “The geometry of n dimensions is a real object, nobody doubts that nowadays. Figures in hyperspace are as susceptible to precise definition as those in ordinary space, and even if we cannot represent them, we can still conceive of them and study them. So if the mechanics of more than three dimensions is to be condemned as lacking in object, the same cannot be said of hypergeometry” (Stillwell, 2010).

La Géométrie à n dimensions a un objet réel; personne n’en doute aujourd’hui

Among the multiple great discoveries in Analysis Situs, Poincaré presents the foundations for what would later be called homology theory, a way of associating a sequence of algebraic structures such as abelian groups or modules with other mathematical objects such as topological spaces. He arrives there by setting up a system for computing Betti numbers by assuming that every manifold can be decomposed into cells which are homeomorphic to simplices (essentially tetrahedrons in n-dimensions), reading off linear equations he called homologies and computing the corresponding Betti numbers by linear algebra (Stillwell 2012, p.557). As Scholz (1980) put it: “The first phase of algebraic topology, inaugurated by Poincaré, is characterized by the fact that its algebraic relations and operations always deal with topological objects.”

Using his new homology theory, Poincaré next provides the Poincaré duality theorem for the Betti numbers for an n-dimensional manifold, by considering the dual of the cell decomposition. The duality theorem states that Betti numbers of the same distance from the ‘ends’, i.e. top and bottom dimensions, are equal. In particular, for a 3-manifold the 2-dimensional Betti number equals the 1-dimensional Betti number (Stillwell, 2012).

Later in the same paper, Poincaré also provides a generalization of the Euler polyhedron formula to arbitrary dimensions, and relates it to his homology theory (Stillwell, 2010). He also gives new examples of fundamental groups which establish that π₁ is a stronger invariant than the Betti numbers, by identifying opposite faces of an octahedron with the same Betti numbers as the 3-sphere, but again, a different fundamental group (namely the cyclic group). The take-away from his discoveries is that for 0-, 1- and 2-dimensional manifolds, the Betti numbers suffice to distinguish them between one another, but that for three-dimensional manifolds, the fundamental group becomes important. How important, Poincaré at this point (1895) cannot answer.

In retrospect, as Stillwell (2010) writes, because of Poincaré’s construction of homology theory and the establishment of the fundamental group, Analysis Situs is rightly regarded as the origin of algebraic topology. As for homology theory, the importance of its establishment was that it revealed that there is an algebraic structure giving rise to the Betti numbers (and so the Euler characteristic). The discovery of the fundamental group highlighted the lacking power of Betti numbers as an indicator of manifolds’ properties.

First and second supplements to Analysis Situs (1899–1900)

Analysis Situs, although brilliantly inventive, was provided not without confusion or error (Stillwell, 2010). Exploring in somewhat of a ‘no man’s land’, Poincaré only later came to discover that Betti numbers were only part of the story, an oversight which he would be made aware of three years later with the thesis of a Danish doctoral student, Poul Heegaard.

Poincaré wrote his first supplement, entitled Complément a l’analysis situs (“Supplement to the Analysis Situs”) in 1899. The paper was motivated by the discovery of Heegaard (1898) that Poincaré’s new definition of Betti numbers could be shown to be in conflict with his duality theorem. Heegaard contrasts the example of the real projective space RP³ with the example of the 3-sphere and shows that Poincaré had failed to account for the effects of torsion, “twisting”. After first moving towards a more combinatorial theory of homology, in which manifolds are assumed to have a polyhedral structure, in his supplements (sometimes referred to by Poincaré as complements) Poincaré eventually revises his homology theory to produce torsion numbers in addition to Betti numbers, proved to be invariant using the so-called Hauptvermutung argument (any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent), now known to be false. The inclusion strengthened homology theory to be able to distinguish between manifolds undergoing torsion from one another (including RP³ and the 3-sphere). Poincaré indeed coined the term ‘torsion’ through a discussion of non-orientable surfaces such as the Möbius band (Stillwell, 2010).

In the second supplement paper (1900), Poincaré’s now more robust homology theorem emboldens him to end his paper with the statement:

“Each polyhedron which has all its Betti numbers equal to 1 and all its tables Tq orientable is simply connected, i.e. homeomorphic to a hypersphere”

Effectively conjecturing that

“Any three-manifold with trivial homology is homeomorphic to the 3-sphere”

A first (and incorrect) Poincaré conjecture about the topological properties of three-dimensional bodies undergoing deformation.

Third and fourth supplements to Analysis Situs (1902)

In terms of 3-manifolds and the later Poincaré conjecture, the relevance of the third and fourth supplements (other than expanding on Poincaré’s homology theory and algebraic topology) is in their study of torus bundles, which are shown to arise naturally in the study of algebraic curves, the focus of the third supplement “On certain algebraic surfaces” (1902a) (Stillwell, 2010).

Fifth supplement to Analysis Situs (1904)

Poincaré published his fifth and final supplement to his 1895 Analysis situs paper in 1904. The paper regards three-dimensional manifolds (such as the glome from our introduction) and is entitled Cinquième complément à l’analysis situs. The paper is now known for Poincaré’s study of whether 3-dimensional manifolds can be described by the same distinguishing feature as 2-dimensional manifolds, namely that every simple closed curve in the sphere can be deformed continuously to a point without leaving the sphere. As Poincaré himself writes in his introduction:

This time I confine myself to the study of certain three-dimensional manifolds, but the methods used without doubt are of more general applicability. I shall devote considerable space to certain properties of closed curves which can be traced on closed surfaces in ordinary space.

The paper sets out, not with an investigation of simply connected, closed 3-manifolds, but rather, with a study of the differences between homology and the more extensive homotopy theory, in the case of curves on surfaces via the study of fundamental groups (Stillwell, 2012). The result is an interesting geometric algorithm to decide whether a curve on a surface is homotopic to a simple curve, the first known case of applying geometrization to topology, which would later would inspire the work of Dehn (1910; 1912a; 1912b; 1922), Nielsen (1927; 1929; 1932), and Thurston (1979). The paper goes on to investigate various other properties of curves on surfaces and their role in constructing three-dimensional manifolds (Stillwell, 2010).

In the final pages of the paper, Poincaré’s investigating leads to a new discovery, the homology sphere called the ‘Poincaré homology sphere’, a 3-manifold with the same homology as the 3-sphere, but a different fundamental group. An existence proof, the discovery immediately rejects Poincaré’s initial conjecture about the relationship between 3-manifolds and 3-spheres from 1902. He constructs the object by joining two “filled” bodies of genus 2 (so-called handlebodies) so that certain carefully chosen curves on one of them become identified with curves with very particular properties on the other. From this, he obtains a three-dimensional manifold whose fundamental group can be written down simply in terms of its generators and relations. “By some miracle” (Stillwell, 2012), the fundamental group of the Poincaré homology sphere turns out to have generators a and b whose relationship is:

a⁴ba⁻¹b = b⁻²a⁻¹ba⁻¹ = 1

A seemingly utterly asymmetric object, the homology sphere is in fact exceptionally symmetric and easily shown to be nontrivial (of dimension 2 or higher) by setting (a⁻¹b)² = 1, which maps it to the 60-element icosahedral group a⁵ = b³ = (a⁻¹b)² = 1. Simultaneously, by allowing the generators to commute, one finds that the object collapses to a single element, 1, showing that the homology of the manifold must be trivial. Hence, the homology sphere is in fact a closed three-dimensional manifold with trivial homology but a non-trivial fundamental group. The discovery of this certainty re-demonstrated for Poincaré the power of his fundamental group over homology for distinguishing 3-manifolds, and put a final nail in the coffin for his first and flawed conjecture about the usefulness of homology in demonstrating that all 3-manifolds are homeomorphic to the 3-sphere.

Poincaré ends the fifth supplement to his Analysis Situs with a question:

Is it possible for the fundamental group of a manifold to reduce to the identity without the manifold being simply connected?

..and an obfuscation:

“This question would carry us too far away.”