Introduction

Understanding the universe's shape, both in its past and future, presents a significant empirical challenge. Albert Einstein contributed to this understanding by illustrating how matter and energy (three-dimensional entities) interact with a four-dimensional phenomenon: time. This interaction suggests that spacetime can warp in response to mass and energy. We exist within a four-dimensional universe subject to various deformations—stretching, twisting, and bending—which leads us to the contributions of Henri Poincaré and the development of topology.

To simplify, a circle on a plane is the one-dimensional boundary of a two-dimensional disk. Similarly, the two-dimensional surface of a three-dimensional ball is known as a sphere. However, as we ascend dimensions, our intuitive grasp falters. The three-dimensional boundary of a four-dimensional sphere, known as a glome or 3-sphere, exists mathematically but is challenging to visualize.

In mathematics, these objects—circle, sphere, and glome—are categorized as 1-, 2-, and 3-spheres, respectively. These n-spheres generalize the concept of a sphere to arbitrary dimensions, treated as n-dimensional manifolds resembling Euclidean space at each point.

Author Sylvia Nasar, in her book "A Beautiful Mind," offers an illustrative perspective on manifolds:

Imagine being shrunk to the size of a pinhead, sitting on a doughnut's surface. It appears flat, yet if you were on a three-dimensional manifold, your immediate surroundings would resemble the interior of a ball. Thus, how an object looks from afar may differ significantly from its local appearance.

Before Topology (1752–1895)

In his book "Papers on Topology: Analysis Situs and Its Five Supplements" (2010), John Stillwell asserts that prior to Poincaré, only one universal topological concept existed: the Euler characteristic, articulated by Euler's formula V - E + F = ?, where V represents vertices, E edges, and F faces. Both spheres and convex polyhedra, like Platonic solids, share an Euler characteristic of two. In 1863, Möbius demonstrated that all closed surfaces in R³—orientable surfaces—can be classified by their Euler characteristic.

Stillwell also notes the contributions of mathematicians like Gauss (1827) and Bonnet (1848) concerning surface curvature, alongside Riemann's (1851) exploration of algebraic curves. Cayley (1859) further studied surface "pits, peaks, and passes," but significant progress in studying these concepts in arbitrary dimensions only came with Enrico Betti (1871).

Betti introduced what would later be known as Betti numbers P?, P?, P?, etc., defining them as the number of k-dimensional holes on a topological surface. For instance, a torus has Betti numbers P? = 1, P? = 2, P? = 1, indicating one connected surface component with two circular holes enclosing a cavity. The second Betti number, the genus, corresponds to the Euler characteristic. John Milnor's statement about the Poincaré Conjecture for the Clay Mathematics Institute’s Millennium Prize highlights the understanding of surfaces in the 19th century and their defined genus.

Milnor concludes with sketches of surfaces of genus 0, 1, and 2.

Prior to Poincaré's contributions, as Milnor and Stillwell note, only the theory of closed surfaces (2-manifolds) was well-defined, characterized as compact and boundary-less. The classification theorem states that any connected closed surface is homeomorphic to one of three families: 1. The sphere; 2. The connected sum of g tori (g ? 1); 3. The connected sum of k real projective planes (k ? 1). Milnor emphasizes that the corresponding question in higher dimensions is significantly more complex.

Henri Poincaré (1854–1912)

Henri Poincaré was born on April 29, 1854, in Nancy, France. His father was a professor of medicine, and his mother homeschooled him, particularly during his childhood illness. At eight, he entered the Lycée, where he excelled in all subjects and received accolades for his mathematical talent.

In 1871, Poincaré graduated with a B.Sc. in letters and science and served in the Franco-Prussian War. He studied mathematics at École Polytechnique and published his first paper at 22, demonstrating properties of surface indicators. Simultaneously, he pursued mining engineering at École des Mines and graduated in 1879.

Despite earning a doctorate in mathematics, he continued as a mining engineer while teaching at his alma mater and developing a new field known as the qualitative theory of differential equations. Poincaré's explorations spanned various mathematical domains, earning him the title "The Last Universalist."

Poincaré’s Work on Topology

In the 1890s, Poincaré initiated foundational work in topology and algebraic topology with his publication "Analysis Situs" (1895) and five subsequent supplements. The term 'Analysis Situs' derives from Gottfried Leibniz's writings over 150 years earlier.

Poincaré's motivation for establishing topology focused on the need for mathematicians to assess the properties of geometric figures undergoing deformation, coining the term "rubber-sheet geometry."

> “It has often been said that Geometry is the art of reasoning well about poorly drawn figures; these figures, to avoid deception, must satisfy certain conditions; proportions may be grossly altered, but the relative positions of the various parts must not be disturbed.”

This aligns with the modern definition of topology:

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

On Analysis Situs (1892)

In his initial exploration of topology, Poincaré sought to motivate "Analysis Situs" by questioning whether Betti numbers alone could determine a manifold's topological classification. He introduced the concept of the fundamental group, a crucial topological invariant.

> Begin with a space (e.g., a surface) and a point within it, considering all loops that start and end at this point. Two loops are equivalent if one can be deformed into the other without breaking.

He illustrated that certain three-dimensional manifolds share the same Betti numbers but belong to different fundamental groups, arguing that Betti numbers alone cannot differentiate three-dimensional manifolds.

Analysis Situs (1895)

The Poincaré conjecture (1904) had not yet formed in 1895, as he likely assumed that simply-connected n-dimensional closed manifolds would be homeomorphic to the n-sphere. However, "Analysis Situs" aimed to refine and supplement Betti numbers, establishing a more robust foundation.

Poincaré proposed that the geometry of n dimensions is a legitimate object worthy of precise definition, even if visualization proves challenging. Among his groundbreaking discoveries was the foundation for what would become homology theory, linking algebraic structures with topological spaces.

Using his homology theory, Poincaré presented the Poincaré duality theorem, establishing that Betti numbers at equal distances from the top and bottom dimensions are equal. Notably, for a 3-manifold, the 2-dimensional Betti number equals the 1-dimensional Betti number.

Additionally, he generalized the Euler polyhedron formula to arbitrary dimensions, relating it to his homology theory. He also identified examples of fundamental groups, asserting their significance over Betti numbers for distinguishing manifolds.

First and Second Supplements to Analysis Situs (1899–1900)

Although "Analysis Situs" was groundbreaking, it was not without errors. Poincaré's first supplement emerged in 1899, prompted by Poul Heegaard's discovery that his definition of Betti numbers conflicted with his duality theorem.

Poincaré's second supplement in 1900 further solidified his homology theorem, leading him to conjecture that any three-manifold with trivial homology is homeomorphic to the 3-sphere.

Third and Fourth Supplements to Analysis Situs (1902)

The relevance of the third and fourth supplements lies in their exploration of torus bundles, which naturally arise in algebraic curve studies.

Fifth Supplement to Analysis Situs (1904)

Poincaré's fifth and final supplement focused on three-dimensional manifolds, investigating whether they share distinguishing features with their two-dimensional counterparts. He explored the differences between homology and homotopy theories, resulting in new insights into curve properties.

In the concluding pages, Poincaré discovered the Poincaré homology sphere, a 3-manifold with the same homology as the 3-sphere but a different fundamental group. This contradicted his earlier conjecture about the relationship between 3-manifolds and 3-spheres.

Poincaré concluded this supplement with a thought-provoking question:

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

The Poincaré Conjecture (1904)

The conjecture states that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. In contemporary terms, it asserts that if every simple closed curve within the manifold can be continuously deformed to a point, then the manifold is homeomorphic to the sphere S³.

In essence, the conjecture implies that if the universe is a simply connected, closed 3-manifold, it is homeomorphic to a sphere. This connection suggests that while the universe may resemble a 3-torus, it cannot transform into a 3-sphere or vice versa.

This overview of the Poincaré conjecture's history is primarily derived from John Stillwell's "Papers on Topology: Analysis Situs and Its Five Supplements" (2010), with simplifications for clarity. For those interested in delving deeper, Stillwell's paper "Poincaré and the Early History of 3-manifolds" (2012) is particularly recommended.

This essay is part of a series on mathematical topics published in Cantor’s Paradise, a weekly Medium publication. Thank you for reading!