Atlas / Mathematics / The Geometry Thread

Field · Emerged 1895 (roots from 1750)

Algebraic Topology

Which features of a shape survive any amount of stretching, and how can algebra detect them?

6 chapters4 min read7 turning points 1 open problem

Branched from: Euclidean Geometry
Branched into: Algebraic Geometry + Geometric Topology
Figures: Leonhard Euler, Simon Lhuilier, Henri Poincaré, L. E. J. Brouwer, Emmy Noether, Heinz Hopf, Samuel Eilenberg, Norman Steenrod

In brief

Algebraic topology studies the features of a shape that survive any stretching or bending without tearing: how many pieces it has, how many holes, how loops can wind around it. It detects them by attaching algebraic objects (numbers, groups) to each space.

The logic is simple and powerful. Deforming a shape never changes its algebra, so if two spaces have different algebra, no deformation can turn one into the other. It began with Euler counting the corners, edges and faces of solids.

Key ideas

Euler characteristicEnters 1750 – 1758: $\chi = V - E + F$ for a surface cut into faces, and an alternating count of cells in any dimension. It is 2 for a sphere, 0 for a torus, and $2 - 2g$ for a surface with $g$ holes.
Fundamental groupEnters 1895: The loops in a space based at a point, where two loops count as the same if one can be deformed into the other. A sphere's is trivial. A torus's records loops around each of its two circles.
HomologyEnters c. 1925: A sequence of groups $H_0, H_1, H_2, \ldots$ that count a space's holes of each dimension: components, tunnels, enclosed voids. Computable, and unchanged by deformation.
HomotopyEnters 1931: A continuous deformation of one map into another. Two spaces are homotopy equivalent if each can be squeezed onto the other. Most invariants in the field only see a space up to homotopy.
Fixed pointEnters 1911 – 1912: A point a map leaves where it is. Topology can guarantee one exists without finding it, as Brouwer's theorem does for any continuous map of a disc to itself.

Draws on other domains

↙ Physics
Topological Matter
A periodic table of topological matter

Chapter I

A Formula That Ignored Measurement

Euclid's geometry is about lengths, angles and areas. In 1750 Euler noticed something about solids that involved none of them. Count the vertices $V$ , edges $E$ and faces $F$ of any convex polyhedron (a cube, a tetrahedron, the panels of a football), and

V - E + F = 2 .

A cube gives $8 - 12 + 6$ . An icosahedron gives $12 - 30 + 20$ . Squash the solid, bend its faces, stretch it into any shape, and the count never changes. It was a fact about how the pieces are connected, not how large they are. Leibniz had once wished for an analysis situs, a geometry of position rather than magnitude. Euler, who had already solved the bridges of Königsberg by ignoring distances, supplied its first real theorem.

That is why the field's parent is Euclidean geometry. The formula was born as a theorem about Euclid's solids, and it had to be pried loose from them.

Chapter II

The Counterexamples

The formula did not stay simple. In 1813 Simon Lhuilier produced solids where it fails: a polyhedral picture frame gives $V - E + F = 0$ , and each extra tunnel lowers the count by two. Lakatos later made this episode famous as a case study in how mathematics grows. Every counterexample was met either by narrowing the definition of "polyhedron" or by realising that the formula measured something new.

The second response won. $V - E + F$ measures the number of holes, the genus, and it is a property of the surface, not of any particular way of cutting it into faces. By the mid-nineteenth century Riemann was using the same kind of count to classify surfaces, and Möbius and Jordan were classifying closed surfaces completely by it.

Chapter III

Poincaré Attaches Algebra to Shape

Higher dimensions needed more than one number. Henri Poincaré's Analysis Situs (1895) and its five supplements supplied two new tools. One was the fundamental group, which records how loops in a space can and cannot be deformed into one another. The other was an early form of homology, which counts holes of every dimension through Betti numbers. Topology became the study of the algebraic shadows that shapes cast.

His work was brilliant and famously loose. Proofs depended on intuitions that later generations had to rebuild. L. E. J. Brouwer put it on a rigorous footing around 1911–12 with the idea of the degree of a map. With it he proved two landmark results: every continuous map of a ball to itself has a fixed point, and spaces of different dimensions cannot be topologically the same.

Chapter IV

From Numbers to Groups

The decisive shift came from algebra. Around 1925 Emmy Noether, in her Göttingen lectures and in conversation with visiting topologists, pointed out that Betti numbers are only the sizes of richer objects: homology groups. A group can be mapped to another group, and a continuous map between spaces induces exactly such a map between their homology groups. Topology became functorial. Spaces and maps on one side are matched faithfully by algebra on the other, so a question about shape can be answered by computing with algebra.

The rest of the century built on that insight: cohomology, homotopy groups, fibrations and spectral sequences. By 1950 the field had its own axioms, from Samuel Eilenberg and Norman Steenrod, and had spread across mathematics. And it fed directly into geometric topology. Poincaré's conjecture was stated in terms of his own fundamental group.

Chapter V

A Closer Look: Counting the Holes in a Doughnut

Algebraic topology turns "how many holes?" into a computation. Start with the Euler characteristic. Cut a surface into vertices, edges and faces in any way you like and compute $\chi = V - E + F$ . The answer never depends on the cutting.

For a torus, take a single square and glue its opposite edges together: left to right makes a tube, then top to bottom closes the tube into a doughnut. After gluing, all four corners of the square are the same point, the two horizontal edges are one edge, and the two vertical edges are one edge. So

\chi(\text{torus}) = V - E + F = 1 - 2 + 1 = 0 .

A sphere, cut like a cube, gives $8 - 12 + 6 = 2$ . In general a surface with $g$ holes has $\chi = 2 - 2g$ , so the Euler characteristic detects the holes.

Homology records more. The first homology group of the torus is $H_1 = \mathbb{Z}^2$ : two independent loops, one around the tube and one through the hole, that cannot be deformed into each other or shrunk away. For the sphere, $H_1 = 0$ : every loop shrinks. Because deformation never changes these groups, they prove at once that no stretching turns a doughnut into a ball, the kind of impossibility statement the field exists to make. And Euler's number reappears: $\chi$ is the alternating sum of the ranks of the homology groups, here $1 - 2 + 1$ .

Chapter VI

Wrapping Spheres Around Spheres

The simplest-looking questions in the field are still unanswered. How many ways can one sphere be wrapped around another? Heinz Hopf's discovery in 1931 that a 3-sphere wraps nontrivially around a 2-sphere, in infinitely many distinct ways, showed that these homotopy groups of spheres are wild. Computing them has driven the field's most powerful machinery for more than ninety years. Some of the answers now come from computer-assisted calculation, and the table still has no visible end.

Turning points7

1750 – 1758PROOFContested
Euler's polyhedron formula
In a 1750 letter to Goldbach, and in papers published in 1758, Euler observes that for every convex polyhedron the numbers of vertices, edges and faces satisfy $V - E + F = 2$ . It is a fact about solids from Euclid's world. Yet it ignores every length and angle and depends only on how the pieces connect: the first sign of a geometry without measurement.
What is disputed
Two things are disputed. First, priority: a manuscript by Descartes from around 1630, known only through a copy Leibniz made and rediscovered in 1860, contains an equivalent relation about the angles of a polyhedron. Historians disagree about whether Descartes had the formula in any meaningful sense. Second, the proof: Euler's argument had gaps, and Legendre (1794) is usually credited with the first rigorous proof for convex polyhedra.
Leonhard Euler
› Sources (2)
- Richeson, D. S. (2008). Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
- Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
1813CRISIS
Polyhedra with tunnels break Euler's formula
Simon Lhuilier catalogues polyhedra for which $V - E + F \neq 2$ : a picture frame, a solid with a tunnel through it, a cube with a smaller cube inside. The formula was not wrong so much as incomplete. The number $V - E + F$ was measuring something about the solid, the number of holes, that geometry had no word for yet.
Simon Lhuilier
› Sources (2)
- Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
- Richeson, D. S. (2008). Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
1895REFORMULATION
Poincaré's Analysis Situs
Henri Poincaré publishes a long memoir, followed by five supplements, that introduces the fundamental group and a precursor of homology, along with Betti numbers generalising Euler's count to any dimension. For the first time, algebraic objects are attached to spaces in order to tell them apart.
Henri Poincaré
› Sources (1)
- Poincaré, H. (1895). Analysis situs. Journal de l'École Polytechnique (2) 1: 1–123.
1911 – 1912PROOF
Brouwer's fixed-point and degree theorems
L. E. J. Brouwer proves that every continuous map of a closed ball to itself has a point that stays put, and that dimension is a topological invariant: $\mathbb{R}^m$ and $\mathbb{R}^n$ cannot be matched up continuously both ways unless $m = n$ . Topology now had theorems as sharp as any in geometry, proved with invariants that counted how many times a map wraps a space around another.
L. E. J. Brouwer
› Sources (1)
- Brouwer, L. E. J. (1912). Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen 71: 97–115.
c. 1925REFORMULATION
Homology becomes a group, not a count
In lectures and conversations at Göttingen, Emmy Noether urges topologists to treat Betti numbers as shadows of algebraic groups, the homology groups. Heinz Hopf, Leopold Vietoris and Walther Mayer take up the idea within a few years. Numbers become structures that maps between spaces act on, and algebraic topology becomes algebra in earnest.
Emmy Noether
› Sources (1)
- Dieudonné, J. (1989). A History of Algebraic and Differential Topology, 1900–1960. Birkhäuser.
1931PROOF
The Hopf fibration wraps a 3-sphere around a 2-sphere
Heinz Hopf finds a map from the 3-sphere onto the ordinary 2-sphere that cannot be deformed to a constant. Above every point sits a whole circle, and any two of those circles are linked. Higher homotopy groups of spheres turned out to be nonzero, and the long problem of computing them began.
Heinz Hopf
› Sources (1)
- Hopf, H. (1931). Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Mathematische Annalen 104: 637–665.
1945 – 1952REFORMULATION
Homology is given axioms
Samuel Eilenberg and Norman Steenrod characterise homology theories by a short list of axioms, announced in 1945 and developed in their 1952 book. The many competing constructions of homology turn out to agree wherever the axioms hold, giving the field a foundation as clean as Hilbert's for geometry.
Samuel Eilenberg, Norman Steenrod
› Sources (2)
- Eilenberg, S. & Steenrod, N. E. (1945). Axiomatic approach to homology theory. Proceedings of the National Academy of Sciences 31(4): 117–120.
- Eilenberg, S. & Steenrod, N. (1952). Foundations of Algebraic Topology. Princeton University Press.

Applications

Where it is used

Data science
Topological data analysis
Persistent homology looks for loops, voids and clusters in a cloud of data points at every scale at once, and keeps the features that persist. It finds structure that clustering and linear methods miss, in data from protein shapes to neuron activity to materials.
› Sources (1)
- Carlsson, G. (2009). Topology and data. Bulletin of the AMS 46(2): 255–308.
Engineering
Proving sensor coverage without coordinates
Scatter cheap sensors that know only which neighbours they can hear, not where they are. De Silva and Ghrist showed that a homology computation on that communication network can certify that the sensors cover a region with no holes.
› Sources (1)
- de Silva, V. & Ghrist, R. (2007). Coverage in sensor networks via persistent homology. Algebraic & Geometric Topology 7: 339–358.
Economics
Why equilibria exist
John Nash's 1950 proof that every finite game has an equilibrium is a fixed-point argument. His one-page paper used Kakutani's extension of Brouwer's theorem, and a 1951 version used Brouwer's directly. Existence theorems for market equilibria in economics follow the same topological route.
› Sources (1)
- Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences 36(1): 48–49.
Condensed matter physics↗ Physics · Topological Matter
Topological phases of matter
Some materials conduct electricity in steps fixed by integers that no impurity can change. Thouless and collaborators showed in 1982 that those integers are topological invariants of the electrons' quantum states. The idea grew into the field of topological insulators, and it earned a share of the 2016 Nobel Prize in Physics.
› Sources (1)
- Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. (1982). Quantized Hall conductance in a two-dimensional periodic potential. Physical Review Letters 49: 405–408.

Open problems

Where the map runs out

Open

The homotopy groups of spheres

Open. Stable groups are computed through about dimension 90 as of 2026, with no general formula.

In how many essentially different ways can a sphere of one dimension be wrapped around a sphere of another? The answers, the homotopy groups $\pi_{n+k}(S^n)$ , are known for small cases and follow no visible pattern. Wrapping a 3-sphere around a 2-sphere, for instance, already gives infinitely many different ways (Hopf, 1931).

Why it is hard: Each new group is extracted from spectral sequences: layered algebraic machines whose outputs depend on hidden "differentials" that must be determined one at a time, often by separate ingenious arguments. Recent progress (Isaksen, Wang and Xu) used computer calculation alongside motivic homotopy theory to reach dimension 90, but each step outwards is harder than the last.
What resolving it unlocks: These groups control how spaces can be glued together. They underlie the classification of manifolds and of exotic smooth structures on spheres, so every new calculation feeds directly into geometric topology.

› Sources (2)

Ravenel, D. C. (1986). Complex Cobordism and Stable Homotopy Groups of Spheres. Academic Press.
Isaksen, D. C., Wang, G. & Xu, Z. (2023). Stable homotopy groups of spheres: from dimension 0 to 90. Publications mathématiques de l'IHÉS 137.

Recently resolved

The Kervaire invariant one problem

Hill, Hopkins and Ravenel ruled out every dimension above 126 (announced 2009, published 2016). In 2024 Lin, Wang and Xu showed that dimension 126 does have such manifolds (preprint). So they exist in exactly the dimensions 2, 6, 14, 30, 62 and 126.

In which dimensions do manifolds exist that carry a certain invariant, the Kervaire invariant, equal to one? Such manifolds were known in dimensions 2, 6, 14, 30 and 62. The question was whether the pattern continues.

Why it is hard: The problem sat in the part of stable homotopy theory where computations are hardest. The 2009 proof needed a new equivariant version of homotopy theory built for the purpose. The dimension-126 case then required large-scale machine-assisted computation of spectral sequences.
What resolving it unlocks: It settles which dimensions admit certain framed manifolds, and so which spheres can carry exotic smooth structures of a particular kind. The equivariant methods it forced into existence now drive a large part of homotopy theory.