Topology.

OpeningWhat topology is.

Geometry minus distance. The study of properties that survive continuous deformation.

Stretch a doughnut into a coffee mug; topologically, you have changed nothing. The single hole survives. Cut the doughnut and you have changed something — connectedness has broken. Topology asks which properties of a shape are genuine in this sense and which are merely metric accidents.

The discipline organises around three large camps. Point-set topology defines continuity and limits without coordinates. Algebraic topology attaches groups and rings to spaces so that holes become numbers. Differential topology studies smooth manifolds — surfaces good enough to do calculus on.

This deck moves through the founders, the canonical objects, the four-color and Poincaré problems, and the strange afterlife of topology in physics, biology, and machine learning.

Chapter IEuler's bridges.

The founding problem. The Prussian city of Königsberg straddled the Pregel river, with seven bridges connecting two islands and the two banks. Could a citizen walk a route that crossed every bridge exactly once?

Leonhard Euler answered in 1736. No. He stripped the city to a graph — four nodes, seven edges — and observed that any such walk must enter and leave each node the same number of times, except possibly its endpoints. Königsberg had four nodes of odd degree. No walk of the requested type can exist on any graph with more than two odd-degree vertices.

The result is trivial. The method is not. Euler had performed the first piece of mathematics in which the precise positions of objects did not matter — only their pattern of connection. Analysis situs, "analysis of position," was the name he and Leibniz gave it. Topology is what we call it now.

Chapter IIThe Euler characteristic.

Take any convex polyhedron. Count the vertices V, the edges E, and the faces F. Then V − E + F = 2. The cube: 8 − 12 + 6 = 2. The tetrahedron: 4 − 6 + 4 = 2. The dodecahedron: 20 − 30 + 12 = 2.

Euler announced this in 1750. Cauchy and Legendre completed the proofs. The number is a topological invariant — it depends only on the surface, not the way it is divided into faces. The sphere has Euler characteristic 2. The torus, no matter how triangulated, gives 0. A two-holed surface gives −2.

The genus g (number of holes) is captured by χ = 2 − 2g. The Euler characteristic was the first invariant that distinguished surfaces from one another by a single number, and it remains the first invariant a student of topology meets.

Chapter IIIThe Möbius strip.

Take a paper strip, give one end a half-twist, glue it to the other. You now hold a surface with one side. An ant walking along the centre returns to its starting point upside-down. Cut along the centre and you do not get two strips — you get one, twice as long.

August Möbius and Johann Listing independently discovered this in 1858. Listing coined the word Topologie in 1847; Möbius's name attached to the strip.

The strip is the simplest non-orientable surface. Orientability — whether a consistent notion of "left" and "right" can be defined globally — is a topological property invisible to local geometry. Two surfaces with the same curvature can disagree on it.

The strip survives in pop culture as the recycling logo and as an Escher drawing of ants. It survives in mathematics as the prototype of every non-orientable manifold.

Chapter IVThe Klein bottle.

Take a Möbius strip and glue its single edge to itself, smoothly. The result is a closed surface with no boundary, no inside, and no outside. Felix Klein described it in 1882.

The Klein bottle cannot be embedded in three-dimensional space without self-intersection. The familiar glass models — narrow neck threading through the side and re-entering the base — are honest immersions of an object that lives properly in four dimensions.

Cut a Klein bottle along the right curve and it falls into two Möbius strips. Cut a Möbius strip along its centre and it becomes a single two-sided strip with two full twists. These dissection surprises are not parlour tricks; they encode the fact that surfaces can be classified by genus and orientability alone, a theorem proved by Möbius, Jordan, and Dehn between 1860 and 1907.

Chapter VPoint-set topology.

By 1900 it was clear that topology needed foundations free of pictures. Felix Hausdorff's Grundzüge der Mengenlehre (1914) gave them.

A topological space is a set X together with a collection of subsets — the open sets — closed under arbitrary unions and finite intersections, and containing X and the empty set. Continuity, convergence, compactness, connectedness — every classical concept reduces to statements about open sets.

The reward is generality. The same theorems describe Euclidean spaces, function spaces of infinite dimension, the Zariski topology of algebraic geometry, and the étale topology of arithmetic. Hausdorff's axioms are now the universal language of every part of mathematics that says the word "near."

Chapter VIOpen and closed sets.

An open set in the real line is one for which every point has some interval entirely inside the set. (0,1) is open. [0,1] is closed. Closed = complement of open.

Sets can be both — the empty set and the whole space always are. Sets can be neither — the half-open interval [0,1) is. The student's first surprise. The lay intuition that "open" and "closed" are exhaustive opposites is wrong, and the moment that breaks is the moment topology begins.

The open-set definition makes continuity trivial: f is continuous iff the pre-image of every open set is open. No epsilons, no deltas. The price is abstraction; the reward is that the same definition handles every space.

Chapter VIIContinuous maps.

A continuous map between topological spaces preserves nearness in the open-set sense. A homeomorphism is a continuous map with a continuous inverse. Spaces related by a homeomorphism are topologically the same.

The doughnut and the coffee mug are homeomorphic. The sphere and the torus are not — no continuous bijection between them has continuous inverse, because (as we will see) their fundamental groups differ.

Topology is, at its heart, the classification of spaces up to homeomorphism. The classification is solved completely for surfaces, partially for three-manifolds (since Perelman), and wide open in dimensions four and above.

Chapter VIIICompactness.

A space is compact if every open cover has a finite sub-cover. The closed interval [0,1] is compact. The open interval (0,1) is not — the cover by (1/n,1) for n=2,3,… has no finite sub-cover. The whole real line is not.

The notion is harder than open and closed and pays for itself many times over. Compactness implies that continuous functions attain their maxima. It implies that sequences have convergent subsequences (in metric spaces, the Heine-Borel theorem). It is the topological reason the calculus of one variable behaves the way it does.

Tychonoff's theorem (1930) — that arbitrary products of compact spaces are compact — is one of the few statements in mathematics whose truth is equivalent to the axiom of choice.

Chapter IXConnectedness.

A space is connected if it cannot be split into two non-empty open sets. The real line is connected; two disjoint intervals are not. Path-connected means every pair of points can be joined by a continuous path. The two notions agree for "nice" spaces but diverge in pathological ones — the topologist's sine curve is the standard counter-example.

Connectedness is preserved under continuous maps, which is why intermediate-value theorems work — a continuous real-valued function on a connected space takes every value between any two it takes.

The number of connected components is the most basic topological invariant. It corresponds, in algebraic topology, to the zeroth homology group H₀.

Chapter XThe algebraic turn.

By 1900, Henri Poincaré had recognised that topology needed weapons stronger than Euler characteristic. His five papers on Analysis situs (1895–1904) founded algebraic topology: the systematic attachment of algebraic objects (groups, rings, modules) to topological spaces, in a way that homeomorphisms become isomorphisms.

The bargain is profound. Topological problems — does this map exist? are these spaces the same? — become algebraic ones, where calculation is possible. The bargain has been kept for 130 years.

Poincaré introduced fundamental groups and the first version of homology. The 20th century industrialised the subject with cohomology rings (Alexander, Čech), spectral sequences (Leray, Serre), and category theory (Eilenberg-Mac Lane). Modern algebraic topology is one of the densest theories in mathematics.

Chapter XIHomotopy.

Two continuous maps are homotopic if one can be continuously deformed into the other. Two spaces are homotopy-equivalent if maps go between them whose compositions are homotopic to the identity.

Homotopy equivalence is coarser than homeomorphism. A solid disc is homotopy-equivalent to a point. An annulus is homotopy-equivalent to a circle. The information lost is geometric; the information kept is topological.

Most invariants of algebraic topology are homotopy invariants — they cannot tell homotopy-equivalent spaces apart, but they do separate spaces with different homotopy type. Computing the homotopy groups of spheres remains, after a century, one of the hardest problems in mathematics. The seventh homotopy group of the seventh sphere has order 240.

Chapter XIIHomology.

The most computable algebraic invariant. To each space X, homology assigns a sequence of abelian groups H₀(X), H₁(X), H₂(X), … . Roughly: H_n counts the n-dimensional holes.

The torus has H₀ = ℤ (one component), H₁ = ℤ ⊕ ℤ (two independent loops), H₂ = ℤ (one enclosed cavity). The sphere has H₀ = ℤ, H₁ = 0, H₂ = ℤ. The two are not homotopy-equivalent because their first homology groups disagree.

The construction was rebuilt from scratch in the 1930s by Alexander, Čech, and Eilenberg. Singular homology, simplicial homology, Čech cohomology — all turn out to agree on reasonable spaces, by deep theorems. Homology is the workhorse of the subject and the entry point for most computations.

Chapter XIIIThe fundamental group.

Pick a point on a space. Consider all loops based there, with two loops counted equivalent if one can be continuously deformed into the other while keeping the basepoint fixed. The set of equivalence classes forms a group under concatenation. This is π₁(X), the fundamental group.

π₁ of a disc is trivial — every loop contracts. π₁ of a circle is ℤ — loops are classified by how many times they wind. π₁ of a figure eight is the free group on two generators, F₂, the smallest non-abelian fundamental group. The two-holed torus has a more complicated π₁ that nonetheless can be written by generators and relations.

π₁ is the most sensitive of the basic invariants. It tells the sphere from the torus and from every more complicated surface. It is also notoriously hard to compute.

Chapter XIVKnot theory.

A knot is an embedding of a circle into three-dimensional space; two knots are equivalent if one can be deformed into the other without cutting. The unknot is the round circle. The trefoil is the simplest non-trivial knot.

The subject originates with Lord Kelvin's 1867 vortex-atom theory — knotted ether tubes as elements of matter. The physics was wrong; the mathematics was generative. Peter Guthrie Tait tabulated knots through ten crossings by 1885. The classification through 16 crossings (1.7 million prime knots) was completed by Hoste, Thistlethwaite, and Weeks in 1998.

The fundamental questions of knot theory — when are two diagrams the same knot? when is a knot the unknot? — turn out to be hard, deep, and beautiful. Knot equivalence was proved decidable by Haken in 1961. Whether the unknot can be recognised in polynomial time was settled positively only in 2011 (by Lackenby).

Chapter XVReidemeister moves.

Knots are presented by two-dimensional diagrams. Two diagrams represent the same knot if and only if one can be transformed into the other by a finite sequence of three local moves: I (twist or untwist a loop), II (slide a strand under another), III (slide a strand across a crossing).

Kurt Reidemeister proved this in 1927; James Alexander and Garland Briggs independently. The theorem reduced knot theory to a combinatorial game on planar diagrams.

To distinguish two knots, find a quantity preserved by all three moves and compute it differently for each. To prove two knots equivalent, exhibit the moves. The first task is the engine of knot invariants; the second is hard, sometimes impossibly hard.

Chapter XVIKnot invariants.

The history of knot invariants is the history of knot theory. Crossing number is intuitive but hard to compute. Tricolorability distinguishes the trefoil from the unknot in three lines. The Alexander polynomial (1928) was the first polynomial invariant — it does not distinguish a knot from its mirror.

The decisive event of modern knot theory was Vaughan Jones's discovery, in 1984, of a new polynomial invariant arising from operator algebras. The Jones polynomial distinguishes most knots from their mirrors and led directly to Witten's interpretation of knot invariants as Feynman integrals over Chern-Simons theory — work for which Jones, Witten, and Drinfeld received Fields Medals in 1990.

The HOMFLY polynomial generalised both Alexander and Jones. Khovanov homology (2000) categorified Jones into a graded chain complex and revealed structure invisible at the polynomial level.

Chapter XVIIThree-manifolds.

Surfaces (2-manifolds) were classified by 1907 — they are determined by genus and orientability alone. Three-manifolds resisted for a century. The list is infinite and was thought to lack pattern.

William Thurston, in lectures at Princeton in the late 1970s, proposed the geometrisation conjecture: every closed three-manifold can be cut into pieces, each modelled on one of eight homogeneous geometries (spherical, hyperbolic, Euclidean, and five others). The conjecture organised the entire subject.

Thurston received the Fields Medal in 1982 for proofs in major special cases. The full conjecture was proved by Grigori Perelman in 2002–2003 using Ricci flow with surgery, building on a programme of Richard Hamilton. The Poincaré conjecture is the spherical-geometry case and falls out as a corollary.

Chapter XVIIIThe Poincaré conjecture.

Posed by Henri Poincaré in 1904. Statement: every closed, simply connected three-manifold is homeomorphic to the three-sphere. In English: if it has no holes, it is a sphere.

The analogous statement in dimension two — every simply connected closed surface is a sphere — was already classical. The four-dimensional analogue was proved by Michael Freedman in 1982 (Fields Medal). The dimensions five and above were dispatched by Stephen Smale in 1961 (Fields Medal). Dimension three — the case Poincaré stated — held out until 2003.

It was named one of the seven Millennium Prize Problems by the Clay Mathematics Institute in 2000, with a million-dollar bounty. It is the only one solved.

Chapter XIXPerelman's proof.

Three preprints, posted to the arXiv between November 2002 and July 2003 by Grigori Perelman of the Steklov Institute in Saint Petersburg. About 70 pages. The proofs were terse, the citations sparse, the announcements modest.

The technique was Ricci flow with surgery — letting the metric on a manifold evolve like heat, smoothing out irregularities, while cutting away pinch-points before they cause the flow to blow up. Hamilton had proposed the programme in 1982; Perelman supplied the missing analytical estimates and the surgery procedure.

Three independent verification teams (Cao-Zhu; Kleiner-Lott; Morgan-Tian) confirmed the proof through 2006. Perelman was awarded the Fields Medal in 2006. He declined it. He was awarded the Clay Millennium Prize in 2010. He declined that too. He has not published mathematics since.

Chapter XXDifferential topology.

The branch that studies smooth manifolds — spaces locally indistinguishable from Euclidean space, equipped with a notion of differentiation. The objects are the same as in topology; the morphisms are smooth maps; the equivalence is diffeomorphism.

The founding texts are Whitney's embedding theorems (1936) — every smooth n-manifold embeds in ℝ²ⁿ — and Morse theory, which extracts topological information from critical points of smooth functions.

Differential topology is the technical foundation of theoretical physics. General relativity is differential geometry on a 4-manifold. Gauge theory is differential topology with extra fibre-bundle structure. The Yang-Mills equations and the Einstein equations live here.

Chapter XXIManifolds.

The central object. A manifold is a topological space that locally looks like Euclidean space — every point has a neighbourhood homeomorphic to an open subset of ℝⁿ for some fixed n, the dimension.

The Earth's surface is a 2-manifold. The configuration space of a robot arm is a manifold whose dimension equals the number of joints. The state space of a quantum system is an infinite-dimensional manifold (a Hilbert space) plus phase information, making projective space.

Manifolds carry richer structures: topological (continuous), piecewise-linear, smooth, complex analytic, Riemannian (with a metric). Each layer is a category in its own right; the relationships between them produced 20th-century geometry.

Chapter XXIIExotic spheres.

The shock of mid-century topology. The 7-sphere, viewed as a topological manifold, is the same as the round 7-sphere. As a smooth manifold, it is one of twenty-eight different things.

John Milnor proved this in 1956, exhibiting the first exotic sphere — a manifold homeomorphic but not diffeomorphic to S⁷. Smoothness is finer than topology in dimension seven. Milnor and Kervaire (1963) classified smooth structures on S^n for all n ≠ 4: the number is finite and computable.

Dimension four is anomalous. ℝ⁴ admits uncountably many distinct smooth structures (Donaldson, Freedman, mid-1980s) — the only Euclidean space for which this happens. Whether the four-sphere admits exotic smooth structures is the smooth Poincaré conjecture in dimension four, the most famous open problem in topology today.

Chapter XXIIITopological matter.

Topology arrived in physics through three doors. Topological quantum field theory (Witten 1988) packaged invariants of manifolds as path integrals — the Jones polynomial of a knot is a Wilson loop in Chern-Simons theory at level k. Topological insulators (Kane-Mele 2005, Bernevig-Zhang 2006) are materials whose insulating behaviour in the bulk is forced to coexist with conducting states on the boundary, classified by ℤ₂ invariants of the Brillouin zone.

The 2016 Nobel Prize in Physics went to Thouless, Haldane, and Kosterlitz for theoretical discoveries of topological phase transitions. The integer quantum Hall effect — Hall conductance quantised to extreme precision — is now understood as a Chern number of an electronic band.

Topology has become a staple vocabulary in condensed-matter physics. Materials are classified by symmetry-protected topological invariants. Quantum computers based on non-abelian anyons exploit the fact that braiding (a knot-theoretic operation) realises quantum gates.

Chapter XXIVPersistent homology.

The youngest application. Topological data analysis (TDA) treats a finite point cloud as a sample from an unknown topological space and asks what shape it has.

The technique: thicken the points by ε-balls, compute homology of the resulting union, increase ε from zero to infinity, and record the persistence of each topological feature across scales. Long-lived features are signal; short-lived ones are noise. Persistence diagrams were defined by Edelsbrunner, Letscher, and Zomorodian (2002).

The field is now a recognised branch of applied mathematics, with applications in cosmology (the topology of the cosmic web), neuroscience (the structure of neural activity), materials science, and the analysis of high-dimensional embeddings produced by machine-learning models. Carlsson and Mémoli's foundational papers are standard references.

Chapter XXVKnots in DNA.

Bacterial DNA is circular. When the cell replicates, the daughter molecules are linked. The cell unlinks them by enzyme action — topoisomerases cut, pass strands through one another, and reseal. The topology of the chromosome must be reduced to triviality before division.

Mathematicians and biologists collaborated, beginning with De Witt Sumners in the 1980s, to model these enzymes as crossing changes on a tangle. The result is a tangle calculus that predicts which DNA topologies a given enzyme can produce. The match between predicted and observed topologies has been one of the cleanest applications of pure mathematics to biology.

The same tools are used in the analysis of protein folding, where the polymer is a curve in three-space and topological self-entanglement constrains the folding pathway. Several proteins are known to be genuinely knotted; the function, where studied, depends on the knot type.

Chapter XXVIReading list.

• 1934Topologie — the first textbookAlexandroff & Hopf
• 1955The Shape of Space — popular introJeffrey Weeks
• 1965Topology from the Differentiable ViewpointJohn Milnor
• 1972Algebraic Topology — the standard graduate textEdwin Spanier
• 1974Topology — undergraduate canonJames Munkres
• 1976Knots and LinksDale Rolfsen
• 1982The Geometry and Topology of Three-ManifoldsWilliam Thurston
• 1983Differential TopologyHirsch
• 1991Genera of the Arborescent LinksDavid Gabai
• 2002Algebraic Topology — free online, the modern textAllen Hatcher
• 2003The Poincaré Conjecture — popular accountDonal O'Shea
• 2007Manifold Destiny — the Perelman storyNasar & Gruber
• 2009Topology and Its ApplicationsWilliam Basener
• 2010Computational TopologyEdelsbrunner & Harer
• 2017Topological Data Analysis — the surveyCarlsson & Vejdemo-Johansson

Chapter XXVIIWatch & read.

↑ Poincaré, Ricci flow, and the million-dollar problem

More on YouTube

Watch · Neil deGrasse Tyson on the Möbius strip
Watch · Mathologer on continuous deformation

And read

Hatcher's Algebraic Topology, free at the author's Cornell page, is the standard modern textbook. O'Shea's The Poincaré Conjecture is the best non-technical account of the proof. For a single-volume introduction with the right balance of intuition and rigour, Munkres remains unmatched.

Chapter XXVIII2026.

Geometrisation has organised three-manifolds; Khovanov homology has reorganised knot theory; persistent homology has founded an applied subfield. The major frontier is dimension four — both the smooth Poincaré conjecture and the existence/non-existence of exotic ℝ⁴ structures with prescribed properties remain wide open.

Higher category theory (Lurie, Joyal, Toën) has rebuilt algebraic topology on infinity-categorical foundations and revealed deep connections to logic and theoretical computer science. Homotopy type theory (Voevodsky and others, post-2010) suggests that types in dependent type theory are spaces up to homotopy — a programme that may eventually unify topology with the foundations of mathematics.

Applied topology continues to grow into mainstream data science. The textbooks of 2030 will look different from those of 2010.

Chapter XXIXThe constants.

What topology has taught the rest of mathematics across three centuries:

1. Position is information. The bare pattern of connection — independent of distance, angle, or coordinates — already determines a great deal. Euler's bridges, the Euler characteristic, and the fundamental group are all evidence.

2. Local data has global consequences. Manifolds, fibre bundles, and sheaves all formalise this. The Gauss-Bonnet theorem — local curvature integrates to a topological invariant — is the cleanest example.

3. The right level of abstraction is the one that proves theorems. Hausdorff's open-set definition, Eilenberg-Steenrod's homology axioms, and Grothendieck's topoi each replaced a concrete construction with a structural one and immediately generalised the field.

4. Continuity is the universal language. Whenever one says "near" or "approximately" outside numerical analysis, the underlying mathematics is topology.