5.3 Horizons · Homology & Cohomology

The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

— William P. Thurston, On proof and progress in mathematics (1994)

This is the last chapter of the book, and the first chapter of everything that comes after it. You can compute homology and cohomology by hand, and the last two chapters showed you the machine they run on. Now we look outward, at a few of the places that machine has been taken. In one, a sports league has results that look fine for every team against its neighbours, and yet no ranking fits them; the obstruction is a cohomology class. In another, the 3-sphere turns out to be made entirely of circles, every two of them linked once — a fact that homology cannot see at all.

The tour has three kinds of stop. First we go deeper into things you already know: the best representative of a cohomology class (Hodge theory), and two of its uses, ranking teams and finding angles in data. Then we meet relatives of homology that keep some of its rules and break others: generalized cohomology, homotopy groups, sheaf cohomology. Last come a map of the subject and a guide for further study. Not everything here is proved; some of it is only described, and when that happens we say so.

Hodge theory: the best representative

A cohomology class is a whole family of cochains: any cocycle plus any coboundary represents the same class (§4.2 Cohomology Groups). In §4.1 Cochains: Measuring Shapes you saw the staircase cochain that climbs around a loop; adding the gradient of any height function gives another staircase with the same total climb. Is one of them best? Hodge theory says yes, and the best one is something you can draw: the representative that spreads the climb as evenly as possible.

The smooth version

On a closed manifold with a way to measure lengths and angles (a Riemannian metric), every differential form has a size, and forms have an inner product. The Hodge Laplacian $Δ = d d^{*} + d^{*} d$ combines the exterior derivative $d$ of §4.3 Differential Forms with its adjoint $d^{*},$ which goes the other way. A form is harmonic if $Δ ω = 0,$ which turns out to mean that it is both closed ( $d ω = 0)$ and co-closed ( $d^{*} ω = 0).$

On the flat square torus, for example, the harmonic 1-forms are exactly the constant combinations $a d x + b d y :$ one harmonic form for each element of $H_{dR}^{1} (T^{2}) ≅ R^{2} .$ The cohomology class is an abstract equivalence class; the harmonic form is a single, canonical object you can draw. We will not prove Hodge’s theorem in the smooth setting — it needs analysis (elliptic partial differential equations), and Warner’s textbook gives a complete proof (Warner 1983, ch. 6) — but its discrete version is pure linear algebra, and you can prove it yourself.

On a complex manifold the harmonic forms split further, by how many complex directions they measure, into pieces of type $(p, q) .$ When the manifold is cut out by polynomial equations, its subvarieties, the smaller shapes cut out by more equations, give classes of type $(p, p) .$ Whether these are all of them is the most famous open question about cohomology.

Conjecture The Hodge conjecture

Open since 1950 Posed by W. V. D. Hodge, 1950

On a smooth complex projective variety, every rational cohomology class of type $(p, p)$ is a rational combination of the classes of algebraic subvarieties.

The conjecture predicts that a topological count, the classes of a certain type, is accounted for by geometry: every such class is built, up to rational multiples, from honest algebraic shapes. It is a bridge between topology and algebra that nobody knows how to build in general.

What is known. It is one of the seven Millennium Prize Problems of the Clay Mathematics Institute. It holds for classes of degree $2,$ by the Lefschetz $(1, 1)$ theorem of 1924, and hence for all varieties of dimension at most $3;$ with integer instead of rational coefficients it is false (Atiyah and Hirzebruch, 1962). In 2025 Eyal Markman proved it for abelian varieties of dimension $4,$ and in 2026 further cases were announced, some of them by AI systems, a few of which have since been withdrawn. The general conjecture is open.

A way in. On a complex curve, every class of degree $2$ is a multiple of the class of a point, and a point is an algebraic subvariety, so there is nothing to prove. The first real case is degree $2$ on a surface, where Lefschetz’s theorem follows from the long exact sequence of $0 \to Z \to O \to O^{*} \to 0,$ a sequence of sheaves like those of Chapter 4.7.

(Hodge 1950; Deligne 2000; Markman 2025) Status as of October 2026 All open problems

The discrete version is least squares

Work on a simplicial complex $K$ with real coefficients, and think of a 1-cochain as a flow on the edges: a number on each oriented edge, as in §4.1 Cochains: Measuring Shapes. Write $δ_{0} : C^{0} \to C^{1}$ for the gradient (a function on vertices goes to its differences along edges) and $δ_{1} : C^{1} \to C^{2}$ for the curl (a flow goes to its circulation around each triangle). With the standard inner products, their transposes $δ_{0}^{T}$ and $δ_{1}^{T}$ are “divergence” and “spread a circulation around a triangle’s edges”.

(Why exactly when? Because $h^{T} L_{1} h = ∥ δ_{0}^{T} h ∥^{2} + ∥ δ_{1} h ∥^{2},$ a sum of two squares, and it is zero only if both terms are.) Since $δ_{1} δ_{0} = 0$ — “curl of a gradient is zero” — the image of $δ_{0}$ (gradients) and the image of $δ_{1}^{T}$ (curls) are perpendicular, and what is perpendicular to both is exactly the harmonic flows.

This is the Hodge decomposition previewed in §4.1 Cochains: Measuring Shapes, now with its third piece explained: the harmonic flows are exactly the cohomology. Lek-Heng Lim’s survey develops the whole theory in this matrix language, for readers who know linear algebra and graphs (Lim 2020).

Each piece is found by least squares. The gradient part is the gradient closest to $f :$ choose the potential $s$ to minimise $∥ δ_{0} s - f ∥^{2},$ which means solving the normal equations $δ_{0}^{T} δ_{0} s = δ_{0}^{T} f$ — the graph Laplacian of §4.1 Cochains: Measuring Shapes appears. The curl part is found the same way, and the harmonic part is what is left. And if $f$ is a cocycle, its harmonic part is the smallest cocycle in its cohomology class: Hodge’s “best representative” is the one of least energy.

Example A decomposition with clean numbers

Take four vertices $1, 2, 3, 4$ and edges $12, 23, 13, 34, 14,$ and fill in the triangle $123$ but not $134$ — so there is one hole, surrounded by $1 \to 3 \to 4 \to 1 .$ The flow $f = (3, 3, 3, 2, - 2)$ on $(12, 23, 13, 34, 14)$ splits as

part	on 12, 23, 13, 34, 14	what it is
gradient	$(1, 1, 2, - 1, 1)$	differences of the potential $s = (0, 1, 2, 1)$
curl	$(1, 1, - 1, 0, 0)$	one unit circulating around the filled triangle $1 \to 2 \to 3 \to 1$
harmonic	$(1, 1, 2, 3, - 3)$	a flow around the hole: its total around $1 \to 3 \to 4 \to 1$ is $8$

Check the Pythagoras identity: $∥ f ∥^{2} = 9 + 9 + 9 + 4 + 4 = 35,$ and $8 + 3 + 24 = 35 .$ (These numbers are recomputed by the book’s test suite.) The harmonic flow is divergence-free at every vertex and has zero circulation around the filled triangle, but goes around the hole: no local test can detect it.

Gradient + curl + harmonic Click edges to add flow · choose a complex and a starting flow

click adds

Figure 5.3.1 A flow on a triangulated annulus (or a disk with two holes) and its three Hodge parts, drawn as arrows whose thickness shows their size. The gradient part (teal) flows downhill from high potential to low; the curl part (violet) circulates around individual triangles; the harmonic part (rose) circulates around the holes. Click edges to change the flow and watch the energy split

∥ f ∥^{2} = ∥ grad ∥^{2} + ∥ curl ∥^{2} + ∥ harm ∥^{2} .

However you edit, the harmonic part stays in a space of dimension

b_{1} :

one pattern for the annulus, two for the disk with two holes.

Ranking with cohomology: HodgeRank

Here is an application that needs nothing beyond what we just did. Five teams play some games — not every pair meets — and we know the score margins. Who is best? If every result were consistent with some underlying strengths $s_{i}$ (team $j$ beats team $i$ by exactly $s_{j} - s_{i}),$ the margins would form a gradient flow and we could read off the strengths. Real results are never that tidy: Aurora beats Boreal, Boreal beats Cygnus, and then Cygnus beats Aurora.

HodgeRank, introduced by Xiaoye Jiang, Lek-Heng Lim, Yuan Yao and Yinyu Ye, treats the margins as a flow on the graph of games, with every triangle of mutually played games filled in, and takes its Hodge decomposition (Jiang et al. 2011). The gradient part is the best global ranking, found by least squares. What is left over is inconsistency, and it comes in two kinds. The curl part is “locally cyclic”: rock–paper–scissors among three teams that all played each other. The harmonic part is “locally acyclic but globally cyclic”: cycles around longer loops of games, invisible to any check on three teams at a time.

HodgeRank: who is best? Change the score margins · switch what is drawn

Ranking (the gradient part)

1 Cygnus +0.95

2 Aurora +0.85

3 Boreal +0.40

4 Electra −0.78

5 Draco −1.42

How consistent were the results?

explained by the ranking 57%

local cycles (A, B, C) 33%

a long cycle (A, C, D, E) 9%

A − B 2

B − C 1

A − C −1

C − D 3

D − E 0

A − E 1

Each number is a score margin: how many points the first team scored more than the second. $flow (i \to j) = points_{j} - points_{i}$ , so arrows point to the better team.

Figure 5.3.2 Each game’s margin is a number on an edge, drawn as an arrow towards the team that did better. Aurora, Boreal and Cygnus all played each other, so their triangle is filled; the loop Aurora–Cygnus–Draco–Electra has no diagonals, so it surrounds a hole. In the starting season, Cygnus comes first even though it lost to Boreal: a little over half of the “energy” of the results is explained by the ranking, about a third is a local cycle in the triangle, and about a tenth circulates around the long loop. Try the presets: a consistent season is pure gradient, rock–paper–scissors is pure curl, and “round the houses” is pure harmonic.

The harmonic part has a meaning worth pausing on. It is a cohomology class of the comparison graph — a nonzero element of $H^{1}$ — and it is precisely the inconsistency that no amount of local checking will find. Each team’s results look fine against its neighbours; only by going all the way around the loop do you discover that there is no consistent ranking. That is the local-versus-global obstruction of §4.1 Cochains: Measuring Shapes, this time in a sports league.

Circular coordinates from data

Persistent homology (§3.8 Persistent Homology) can tell you that a cloud of data points has a loop in it. It cannot, by itself, tell you where along the loop each point sits. For that you need cohomology — and the reason is one of the prettiest facts in the subject.

Vin de Silva, Dmitriy Morozov and Mikael Vejdemo-Johansson turned this into an algorithm (de Silva et al. 2011). Build a Vietoris–Rips complex of the data at a scale where a persistent loop is alive. Persistent cohomology hands you an integer cocycle $α$ representing it — typically a “fence” of edges, each counting $\pm 1$ for crossing a cut, just like the staircase cochain. A fence gives a terrible coordinate: it jumps by a whole turn across one set of edges and stays constant elsewhere. So smooth it, by exactly the least squares of the previous section: find the real function $f$ on the vertices that makes $α ˉ = α + δ f$ as small as possible. Then $α ˉ$ is the harmonic representative of the class, and the angle of a data point $v$ is $θ (v) = f (v) mod 1.$ Since $α$ takes integer values, $δ θ$ agrees with $α ˉ$ up to whole turns: the angle changes along each edge by the small amount $α ˉ,$ and goes once around in total.

Angles from cohomology Slide the scale · pick another cocycle · new data

scale

r = 0.32

b_{0} = 1, b_{1} = 1

One loop, so

H^{1} ≅ Z

. The rose “fence” is the integer cocycle

α

(11 edges); smoothing it gives every point an angle

θ \in [0, 1)

, shown as colour. The plot on the right is a straight band that wraps once:

θ

recovers the position around the circle.

Scale r (ball radius) 0.32 Show the cocycle α

Figure 5.3.3 Forty noisy points near a loop. The figure starts in the middle of the range of scales where the Rips complex has exactly one loop (

b_{1} = 1);

there, linear algebra finds an integer cocycle — the rose fence, normalised to vanish on a spanning tree — and least squares smooths it into an angle for every point, shown as colour. The plot on the right compares that angle with each point’s true position: a straight band that wraps once. “Pick another cocycle” changes the fence but not the result (beyond a rotation or reflection), because the harmonic representative depends only on the cohomology class.

Generalized cohomology theories

In §5.2 Homological Algebra we met the five Eilenberg–Steenrod axioms and the theorem that they determine homology on cell complexes. Drop the dimension axiom — allow a point to have nonzero groups in degrees other than zero — and you get generalized (co)homology theories. They keep the long exact sequences, Mayer–Vietoris and spectral sequences, but they see different things. Here are the two most famous, followed by the idea that unifies them.

K-theory: counting vector bundles

A vector bundle over a space $X$ is a family of vector spaces, one for each point, varying continuously — like the tangent planes of a surface, or the Möbius band viewed as a family of lines over a circle (§4.8 Curvature and Characteristic Classes). Two bundles can be added (put the fibres side by side), but not subtracted. Topological K-theory, created by Michael Atiyah and Friedrich Hirzebruch in 1959–61 after Grothendieck’s algebraic version (Atiyah & Hirzebruch 1961), does what we do to build the integers from the natural numbers: it allows formal differences $[E] - [F]$ of complex vector bundles, and declares bundles the same if they become isomorphic after adding a trivial bundle (“stable equivalence”). The result is an abelian group $K^{0} (X),$ and it extends to a whole cohomology theory $K^{n}$ (Hatcher 2017).

K-theory violates the dimension axiom in a spectacular, regular way. Bott periodicity, proved by Raoul Bott in 1959 (Bott 1959), says that complex K-theory repeats with period two: $K^{n} (pt)$ is $Z$ for every even $n$ and $0$ for every odd $n .$ Among its triumphs is an eight-page proof, by Frank Adams and Michael Atiyah, of a theorem of Adams about the Hopf invariant (next section) whose first proof, in 1960, had run to eighty-five pages (Adams & Atiyah 1966). Its most quotable consequence, first proved in 1958 by Raoul Bott and John Milnor and by Michel Kervaire, also with Bott periodicity: $R^{n}$ carries a bilinear multiplication without zero divisors, as the real numbers, complex numbers, quaternions and octonions do, only when $n = 1, 2, 4$ or $8$ (Hatcher 2002, §2.B). K-theory is also the language of the Atiyah–Singer index theorem, which computes analytic invariants of differential operators from topology.

A little history Lights out at Warwick

Raoul Bott 1923 – 2005

Friedrich Hirzebruch 1927 – 2012

Michael Atiyah 1929 – 2019

Bott’s doctorate was in electrical network theory, and when the lights failed during a seminar at Warwick in the late 1960s he leapt up: “I’ll fix them — I’m an electrician!” They came on for a few seconds, twice, and the seminar was held by the headlights of cars driven up to the window (O’Connor & Robertson 2006). Hirzebruch had been drafted at fifteen, with his whole school class, to help at an anti-aircraft battery. From 1957 he organised the Arbeitstagung in Bonn, at whose first thirty meetings Atiyah, who grew up in Khartoum and Cairo, lectured thirty-two times. Grothendieck’s lectures at the first one set the two of them on the road to K-theory (O’Connor & Robertson 2009; O’Connor & Robertson 2014), and Atiyah and Bott went on to a decade of joint work on index theory.

Cobordism: homology made of manifolds

When Poincaré invented homology, his cycles were pieces of manifolds, and two cycles were “homologous” when together they formed the boundary of something (§3.1 Cycles and Boundaries; Poincaré (1895)). Cobordism takes that literally. Two closed $n$ -manifolds $M$ and $N$ are cobordant if together they are the boundary of a compact $(n + 1)$ -manifold: $\partial W = M ⊔ N .$ The cobordism classes form a group, and René Thom computed them in 1954 for unoriented manifolds (Thom 1954) — work for which he received the Fields Medal in 1958. Every closed 1-manifold is a union of circles, and each circle bounds a disk, so in dimension 1 everything is cobordant to nothing. In dimension 2, a closed surface bounds a 3-manifold exactly when its Euler characteristic is even: the torus and the Klein bottle bound, but the projective plane $RP^{2}$ (with $χ = 1)$ does not. So the cobordism group of a point in dimension 2 is $Z /2,$ generated by $RP^{2}$ — another failure of the dimension axiom.

The unifying idea: representability and spectra

Here is a deeper reason why cohomology is contravariant. For any abelian group $G$ and any $n \geq 1$ there is a space $K (G, n),$ an Eilenberg–MacLane space, whose only nonzero homotopy group (next section) is $π_{n} = G,$ and for cell complexes $X$ $H^{n} (X; G) ≅ [X, K (G, n)],$ the set of homotopy classes of maps from $X$ into $K (G, n)$ (Hatcher 2002, Thm 4.57). The circle is $K (Z, 1)$ — that was the circular-coordinates fact above — and infinite complex projective space $CP^{\infty}$ is $K (Z, 2),$ which is why the first Chern class of §4.8 Curvature and Characteristic Classes lives in $H^{2} (X; Z) .$ A cohomology class is a map into a special space: maps pull back, so cohomology pulls back.

Edgar Brown proved in 1962 that every generalized cohomology theory on cell complexes is representable in the same way (Brown 1962). The representing spaces come in a sequence $E_{0}, E_{1}, E_{2}, \dots,$ each equivalent to the loop space of the next: a spectrum (Hatcher 2002, §4.E). Ordinary cohomology, K-theory and cobordism are each represented by their own spectrum, and the study of spectra — stable homotopy theory — is one of the most active areas of modern topology. We can only point at the door.

Homotopy groups: the harder cousin

The fundamental group $π_{1}$ of §2.3 Homotopy records loops up to deformation. Replace loops by spheres and you get the homotopy groups: $π_{n} (X)$ is the set of homotopy classes of maps from the $n$ -sphere into $X$ (sending a chosen point to a chosen point), with a group structure defined much as for loops. For $n \geq 2$ these groups are abelian. They are the most natural “higher-dimensional holes” one could imagine — and they are far harder to compute than homology.

For a first taste, compare spheres. Homology is simple: $H_{k} (S^{n})$ is $Z$ for $k = 0, n$ and zero otherwise. Homotopy starts the same way — $π_{k} (S^{n}) = 0$ for $k < n,$ and $π_{n} (S^{n}) ≅ Z,$ detected by degree (§3.5 Maps, Invariance, and First Triumphs); indeed the Hurewicz theorem says that for a simply connected space the first nonzero homotopy group and the first nonzero homology group (in positive degree) occur in the same degree and are isomorphic (Hatcher 2002, Thm 4.32). But then the two part ways completely:

$k$	1	2	3	4	5	6	7	8	9	10
$H_{k} (S^{2})$	$0$	$Z$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$π_{k} (S^{2})$	$0$	$Z$	$Z$	$Z /2$	$Z /2$	$Z /12$	$Z /2$	$Z /2$	$Z /3$	$Z /15$

The 2-sphere has no homology above degree 2, but it has nonzero homotopy groups in infinitely many degrees. Jean-Pierre Serre proved this in 1953, for every simply connected finite cell complex whose homology mod 2 is not that of a point, using the spectral sequences of §5.2 Homological Algebra (Serre 1953b); nobody knows the groups of $S^{2}$ in every degree. The first surprise is $π_{3} (S^{2}) ≅ Z .$ Its generator is a map from the 3-sphere to the 2-sphere that you can actually see.

A little history Hopf’s surprise

Heinz Hopf 1894 – 1971

A lieutenant on the Western Front in the First World War, he spent a fortnight’s leave in 1917 at Erhard Schmidt’s lectures in Breslau, and came away knowing that he wanted to do research; what fascinated him, he wrote, was “the power of the method of the mapping degree”, and the fascination never left him (O’Connor & Robertson 2000). He found his map in 1931, four years before Witold Hurewicz defined the higher homotopy groups in general. Homology sees nothing between $S^{3}$ and $S^{2}$ — $H_{3} (S^{2}) = 0$ — so it would be reasonable to guess that every map from the 3-sphere to the 2-sphere can be squashed to a point. Hopf showed instead that there are infinitely many essentially different such maps, told apart by an integer, his invariant (Hopf 1931). The tool he used was linking, an idea as old as Gauss.

The Hopf fibration

Think of the 3-sphere as the pairs of complex numbers $(z_{1}, z_{2})$ with $∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} = 1,$ and of the 2-sphere as the complex numbers together with a point at infinity. The Hopf map sends $(z_{1}, z_{2})$ to the ratio $z_{1} / z_{2} .$ Multiplying both $z_{1}$ and $z_{2}$ by the same unit complex number $e^{i t}$ does not change the ratio, so the points of $S^{3}$ that map to any one point of $S^{2}$ form a whole circle — a fibre. The 3-sphere is a union of circles, one over each point of the 2-sphere, none of them intersecting (Hatcher 2002, Example 4.45).

The Hopf fibration Drag to rotate · scroll or pinch to zoom · change the number of circles

Latitudes on S² 3

Show the tori they fill

Figure 5.3.4 The fibres of the Hopf map, drawn in ordinary space by stereographic projection from the 3-sphere. Each coloured circle sits over the point of the same colour on the little base sphere: the circles over one circle of latitude fill a torus (switch on “Show the tori”), and the tori are nested. The two white fibres sit over the poles (the white dots): over the north pole, the flat circle at the core of every torus; over the south pole, the vertical axis — a circle through the point at infinity. The gold and teal circles are two fibres over far-apart points, and like every pair of fibres, they are linked exactly once.

Why does this show that $π_{3} (S^{2}) \neq = 0 ?$ If the Hopf map could be deformed to a constant map, one could also deform away the pattern of its fibres; but the preimages of any two points of $S^{2}$ are two circles that link once, and Hopf showed that this linking number — now called the Hopf invariant — cannot change under deformation. The book’s test suite computes the linking number of pairs of these fibres numerically (by Gauss’s double integral) and gets $\pm 1$ every time. A linked pair of circles, a whole space filled with them, and an integer that refuses to change: that is $π_{3} (S^{2}) ≅ Z .$

Sheaf cohomology and the rising sea

In §4.7 Sheaves and Čech Cohomology you met sheaves — data attached to the open sets of a space, with rules for restricting and gluing — and Čech cohomology, which measures how local data fails to glue into global data. In the 1950s this became the language of a revolution in algebraic geometry, the study of shapes defined by polynomial equations.

The tools were those of the last two chapters. Jean-Pierre Serre’s 1955 paper “Faisceaux algébriques cohérents” brought sheaf cohomology into algebraic geometry (Serre 1955). Alexander Grothendieck’s Tôhoku paper of 1957 (Grothendieck 1957) then did two things at once: it set out the axioms of abelian categories (§5.1 Categories and Functors) and defined sheaf cohomology as the derived functors (§5.2 Homological Algebra) of “take global sections”. Global sections are left exact but not exact, and cohomology measures the failure — the same story as Hom and Ext.

Grothendieck and his school went on to build étale cohomology, a cohomology theory for shapes defined by polynomial equations over any field, finite fields included, where ordinary topology makes no sense. With it, Pierre Deligne proved the last of the Weil conjectures in 1974 (Deligne 1974): the number of solutions of polynomial equations over finite fields is governed by “Betti numbers”, exactly as a Lefschetz fixed-point formula would predict. The method was the one Grothendieck described with an image — not cracking a hard problem with a hammer, but surrounding it with theory until it dissolves. In Colin McLarty’s translation, his first analogy for that approach was “immersing the nut in some softening liquid, and why not simply water?”; the second became the epigraph of §5.2 Homological Algebra: the rising sea (McLarty 2007).

Homology at work

Homology was invented to answer questions in pure mathematics. A century later it turns up in laboratories, in robots and in the theory of materials. Here is a quick, honest tour; the map below collects these and more, with references for each.

Physics. The integer quantum Hall effect, in which the electrical conductance of a two-dimensional material comes in exact integer steps, is explained by a Chern number — a characteristic class (§4.8 Curvature and Characteristic Classes) — as shown by Thouless, Kohmoto, Nightingale and den Nijs in 1982 (Thouless et al. 1982). Topological insulators and superconductors are organised by K-theory and Bott periodicity (Kitaev 2009). Topological quantum field theories were axiomatised by Atiyah in 1988 as, in effect, functors from a category of cobordisms to vector spaces (Atiyah 1988); and physicists use cohomology and cobordism invariants to describe the “anomalies” that can obstruct a quantum field theory from being consistently defined.
Data. Persistent homology (§3.8 Persistent Homology) finds shape in point clouds: a Klein bottle in the space of small patches of natural images (Carlsson et al. 2008), rings and cavities in glass (Hiraoka et al. 2016), loops in the family trees of viruses that swap genes (Chan et al. 2013). Sensors that only know which neighbours they can hear can certify, with homology, that a region is fully covered (de Silva & Ghrist 2007).
Neuroscience. The topology of correlations between neurons can reveal geometric structure in their activity (Giusti et al. 2015), and the nerve theorem of §4.7 Sheaves and Čech Cohomology explains how overlapping “place fields” of neurons could encode the shape of an environment.
Robotics. The positions of a robot form a configuration space. Michael Farber’s topological complexity uses cohomology (its cup products, in fact) to bound from below how many separate continuous rules any motion planner must use; a single continuous rule exists only when the configuration space is contractible (Farber 2003).
Knots. Khovanov homology, built in §3.7 Knots and Links, has a graded Euler characteristic equal to the Jones polynomial: the polynomial is a “shadow” of the homology, as $χ$ is a shadow of the Betti numbers (Khovanov 2000). Promoting a number or polynomial to a homology theory in this way is called categorification, and the extra information pays: Khovanov homology recognises the unknot (Kronheimer & Mrowka 2011), and Jacob Rasmussen extracted from it a lower bound for the genus of the surfaces a knot can bound in four-dimensional space, enough to compute the unknotting numbers of all torus knots without gauge theory (Rasmussen 2010). The Alexander polynomial has its own categorification, knot Floer homology, built by Peter Ozsváth and Zoltán Szabó from holomorphic disks; it detects the genus of a knot exactly (Ozsváth & Szabó 2004; Ozsváth & Szabó 2004).

A map of the subject Select a star to read about that field

Homology & cohomology This book

Where this book lives: chains and cochains, cycles modulo boundaries, and the long exact sequences, products and dualities that make them computable. Every other star on this map is reached from here.

Hatcher, Algebraic Topology (free)

This bookPure mathematicsApplicationsPhysics

Figure 5.3.5 Some of the fields that grew out of homology and cohomology, grouped as pure mathematics (violet), applications (teal) and physics (rose), with lines for strong connections. Each star has a short description and references to start from. The map is a sketch, not a census — every field here has neighbours that did not fit.

Explorations

Instead of exercises, three open-ended explorations. Each has a discussion rather than a solution: there is more than one good answer, and the point is to think.

Exercise Rank something you care about

Choose a small league or tournament where not every pair has played (or any set of pairwise preferences: films, foods, chess openings). Pick five teams and six games that fit the pattern of Figure 5.3.2: three teams A, B, C that all played one another, and two more, D and E, with games C–D, D–E and E–A. Enter the score margins, scaled to lie between $- 5$ and $5 .$ How much of the result is explained by a ranking? Is the inconsistency local (curl) or global (harmonic)? What would it mean, practically, if the harmonic part were large?

Solution

A large gradient share means the results are close to consistent with a single strength per team. A large curl share means there are genuine three-way cycles among teams that all played each other — perhaps styles of play that beat each other in a circle. A large harmonic share is the most interesting: the results are consistent on every triangle you can check, yet impossible to explain by a ranking, because the cycle runs around a loop of games with no shortcuts. Two practical readings: the schedule has a “hole” (some teams never meet), so the league cannot be ranked without more games; and the cheapest new game to schedule is one that fills the hole, turning the harmonic part into curl you can see. This is the local-versus-global theme of §4.1 Cochains: Measuring Shapes in a new costume.

Exercise Linked circles everywhere

In Figure 5.3.4, switch on the tori and watch a single torus while you rotate. (a) Convince yourself that each fibre on a torus winds once around in each direction, so that any two fibres on the same torus are linked once. (b) Why are two fibres on different tori also linked? (c) What would it mean for $π_{3} (S^{2})$ if some pair of fibres were unlinked?

Solution

(a) On the torus over a latitude, a fibre is a curve of type $(1, 1) :$ it goes once around the hole and once around the tube. Two disjoint $(1, 1)$ curves on a torus in space link once — you can see one passing through the other’s disk. (b) The tori are nested, and every fibre on an inner torus passes once through the disk bounded by any fibre on an outer torus; equivalently, the linking number is a continuous integer-valued function of the pair of base points, so it is constant, and we just computed it is 1. The extreme cases are instructive: the fibre over the north pole is the central unit circle, and the fibre over the south pole is the vertical axis (a circle through infinity), and these visibly link. (c) The Hopf invariant would be zero, and Hopf’s argument would give no obstruction to deforming the map to a constant; in fact the invariant is 1, so the Hopf map generates $π_{3} (S^{2}) ≅ Z .$ Going further: the same construction with quaternions gives a map $S^{7} \to S^{4}$ whose fibres are 3-spheres, and shows that $π_{7} (S^{4})$ contains a copy of $Z .$

Exercise Design your own homology theory

The Eilenberg–Steenrod axioms leave room for theories in which a point has homology in many degrees. Suppose you wanted a theory whose “cycles” are closed manifolds mapped into a space, and whose “boundaries” are the manifolds that bound. What should the groups of a point be in dimensions 0, 1 and 2? Which of the five axioms would you expect to survive?

Solution

This is (unoriented) cobordism. For a point, a “cycle” of dimension $n$ is just a closed $n$ -manifold, and it is zero when it bounds. In dimension 0 a closed manifold is a finite set of points, which bounds exactly when the number of points is even (pairs of points bound arcs), so the group is $Z /2 .$ In dimension 1 every closed manifold is a union of circles, each bounding a disk, so the group is $0 .$ In dimension 2 it is $Z /2,$ generated by $RP^{2},$ since a closed surface bounds exactly when its Euler characteristic is even. Homotopy invariance, exactness, excision and additivity all survive — Thom’s work shows this is a genuine generalized homology theory — but the dimension axiom fails, because the point has a nonzero group in dimension 2.

Recap This chapter in brief

Hodge theory: each cohomology class has a unique harmonic representative; on a complex, $L_{1} = δ_{0} δ_{0}^{T} + δ_{1}^{T} δ_{1},$ flows split as gradient + curl + harmonic by least squares, and the harmonic flows form a space of dimension $b_{1} .$
HodgeRank ranks from pairwise comparisons and splits inconsistency into local cycles (curl) and global cycles (harmonic).
Circular coordinates: $H^{1} (X; Z) ≅ [X, S^{1}];$ an integer cocycle smoothed by least squares gives every data point an angle.
Generalized cohomology drops the dimension axiom: K-theory (vector bundles, Bott periodicity) and cobordism (manifolds up to bounding); cohomology theories are represented by spaces and spectra, $H^{n} (X; G) ≅ [X, K (G, n)] .$
Homotopy groups are much harder than homology: $π_{3} (S^{2}) ≅ Z,$ generated by the Hopf fibration, whose circle fibres are linked in pairs.
Sheaf cohomology, defined by derived functors in abelian categories, became the language of algebraic geometry (Riemann–Roch, étale cohomology, the Weil conjectures).
Homology is at work in physics (Chern numbers, K-theory of materials, TQFT), data science, neuroscience, robotics and knot theory.

Where to go next: a study guide

Where you go from here depends on what drew you to this book. Three paths, each with a short annotated list. (If you read only one thing next, make it Hatcher’s Algebraic Topology: after this book, its Chapters 2 and 3 will feel like coming home.)

The pure path: topology, geometry, algebra

Read Hatcher to consolidate, then Bott–Tu for the differential-forms viewpoint and spectral sequences, then Milnor–Stasheff for characteristic classes. K-theory and homological algebra open up from there, and category theory runs alongside everything.

Algebraic Topology — Allen Hatcher free

The standard first graduate text, free online. Having read this book you can read Chapters 2 and 3 straight through; Chapter 4 is homotopy theory, including the Hopf fibration and Eilenberg–MacLane spaces.
Differential Forms in Algebraic Topology — Raoul Bott and Loring W. Tu

Springer GTM 82. De Rham cohomology, Čech cohomology, spectral sequences and characteristic classes, all with differential forms — the natural sequel to Part IV of this book.
Characteristic Classes — John Milnor and James Stasheff

Princeton, 1974. Vector bundles and the cohomology classes that measure their twisting; a model of clear writing.
Vector Bundles and K-Theory — Allen Hatcher free

An unfinished but very readable introduction to topological K-theory and Bott periodicity, free online.
K-Theory — Michael Atiyah

The founder’s own lecture notes (1967; reprinted by Addison-Wesley, 1989). Short and direct.
An Introduction to Homological Algebra — Charles A. Weibel

Cambridge University Press, 1994: derived functors, spectral sequences, group cohomology.
The Rising Sea: Foundations of Algebraic Geometry — Ravi Vakil free

Algebraic geometry in Grothendieck’s style, with sheaf cohomology as a main character; free online and named after the image in this chapter.
Category Theory in Context — Emily Riehl free

For the categorical road: universal properties, adjunctions, limits, and the Yoneda lemma.

The applied path: data, networks, computation

Start with Ghrist for breadth and Lim for the linear algebra of cohomology and Hodge theory; then the original papers behind this chapter’s figures, and a computational text for algorithms.

Elementary Applied Topology — Robert Ghrist

Homology, cohomology, sheaves and persistence through applications — sensor networks, robotics, data, economics — with hundreds of figures. Chapters are available on the author’s page.
Hodge Laplacians on Graphs — Lek-Heng Lim free

Cohomology and Hodge theory as “the linear algebra of matrices satisfying AB = 0”, written for readers who know linear algebra and graphs. The perfect companion to the first half of this chapter.
Statistical Ranking and Combinatorial Hodge Theory — Xiaoye Jiang, Lek-Heng Lim, Yuan Yao and Yinyu Ye free

The HodgeRank paper (Mathematical Programming, 2011).
Persistent Cohomology and Circular Coordinates — Vin de Silva, Dmitriy Morozov and Mikael Vejdemo-Johansson free

Where the circular-coordinates figure comes from (Discrete & Computational Geometry, 2011).
A Roadmap for the Computation of Persistent Homology — Nina Otter, Mason A. Porter, Ulrike Tillmann, Peter Grindrod and Heather A. Harrington free

A practical tutorial: from data to barcodes, with software comparisons (EPJ Data Science, 2017).
Computational Topology: An Introduction — Herbert Edelsbrunner and John Harer

American Mathematical Society, 2010. Algorithms for homology and persistence, with proofs.
Sheaf Theory through Examples — Daniel Rosiak free

An open-access, example-first introduction to sheaves and their cohomology (MIT Press, 2022).
Discrete Differential Geometry: An Applied Introduction — Keenan Crane free

Discrete differential forms and the Hodge decomposition on meshes, for graphics and geometry processing.

The physics path: fields, phases, knots

Physicists usually meet cohomology through differential forms and gauge fields. These books build that bridge, and the two papers show where current research has taken it.

Geometry, Topology and Physics — Mikio Nakahara

The standard bridge text for physicists: homology, cohomology, fibre bundles and characteristic classes with physical applications.
The Geometry of Physics — Theodore Frankel

Differential forms, bundles and gauge theory, from a geometer who writes for physicists (Cambridge University Press).
Gauge Fields, Knots and Gravity — John Baez and Javier P. Muniain

A friendly path from Maxwell’s equations through de Rham cohomology to gauge theory and knot invariants (World Scientific, 1994).
Periodic Table for Topological Insulators and Superconductors — Alexei Kitaev free

How K-theory and Bott periodicity organise topological phases of matter (2009). Advanced, but short.
On Khovanov’s Categorification of the Jones Polynomial — Dror Bar-Natan free

The most accessible account of Khovanov homology, with computations (Algebraic & Geometric Topology, 2002).

Watch alongside: companion videos

Some ideas land better when you watch someone draw them. The videos below make good companions to particular chapters. Watch each one after reading its chapter rather than instead of it: videos skip steps that a book takes slowly, and that is part of their charm. They are listed in the order of the book.

§2.1 Spaces and Continuity You Could Have Invented Homology, Part 1: Topology — Boarbarktree Why classifying spaces by listing them is hopeless, and why that leaves one tool: continuous maps. Hence the plan of the series, and of singular homology — probe a space by mapping simple shapes into it.
§2.3 Homotopy Algebraic Topology: Algebraic Invariants of Spaces — Oxford Mathematics (André Henriques) The first hour of a fourth-year Oxford course: the fundamental group as an invariant, and the properties any invariant should have, homotopy invariance above all.
§2.5 Simplicial Complexes You Could Have Invented Homology, Part 2: Some Simple Spaces — Boarbarktree Convex combinations, convex hulls and the standard simplices, built up slowly and animated with care.
§3.1 Cycles and Boundaries You Could Have Invented Homology, Part 3: Boundaries & The Big Idea — Boarbarktree Boundaries and interiors of simplices, then the big idea: place the boundary of a triangle in a space and ask whether the whole triangle can fill it. The series pauses there, just before the definition of homology.
§3.1 Cycles and Boundaries What is…homology intuitively? — VisualMath (Daniel Tubbenhauer) Slides annotated live. A one-dimensional hole is “the number of necklaces you can put it on”, a two-dimensional one “the number of plugs needed to inflate it”; one slide pushes a big loop across triangles until it hugs a small hole.
§3.3 Homology Groups What is algebraic topology? — Aleph 0 Under fifteen minutes with marker pens: the first homology of the torus computed by hand from one vertex, three edges and two triangles, then the worry that leads to singular homology — does the answer depend on how you cut the space? The pinned comment fixes one slip: it is cycles, not chains, that are taken modulo boundaries.
§3.3 Homology Groups Algebraic Topology: Chains, Cycles, and Homology Classes — Oxford Mathematics (André Henriques) The second hour: homology at an intuitive level first, then chains, cycles and homology classes, with many examples drawn on surfaces at the blackboard.
§4.2 Cohomology Groups What, and why, is coHomology — K-Theory Algebra first: the (co)homology of the tiniest complexes — one group, two groups joined by a map, a short exact sequence — read as the failure of exactness, with topology only at the end. A companion to the opening of §5.2 as well.

For a second pass through the whole subject, Daniel Tubbenhauer’s playlist What is…algebraic topology? runs from homotopy to applications in short episodes, each a set of slides annotated live; the slides are free on his website. The episodes that match this book:

“What is…”	read with	slides
simplicial and singular homologyOr: Cycles modulo boundaries	§3.3 Homology Groups §3.5 Maps, Invariance, and First Triumphs	PDF
cellular homologyOr: Winding around	§3.6 Exact Sequences and Mayer–Vietoris	PDF
relative homology; the Mayer–Vietoris sequenceOr: Calculations modulo subspaces; More than the sum of its parts!?	§3.6 Exact Sequences and Mayer–Vietoris	PDF 1,PDF 2
persistent homologyOr: Applications 2 (topology in data analysis)	§3.8 Persistent Homology	PDF
cohomologyOr: Reversing arrows	§4.1 Cochains: Measuring Shapes §4.2 Cohomology Groups	PDF
the universal coefficient theoremOr: Working integrally rocks	§4.2 Cohomology Groups §5.2 Homological Algebra	PDF
the cohomology ring (two videos)Or: Counting intersections; Polynomials, of course	§4.5 The Cup Product	PDF 1,PDF 2
Poincaré duality; Alexander dualityOr: My face is 0-dimensional; Horned spheres!?	§4.6 Poincaré Duality	PDF 1,PDF 2
a (co)homology theoryOr: Shut up and calculate	§5.1 Categories and Functors §5.2 Homological Algebra	PDF
the Künneth formulaOr: Multiplication vs. tensor product	§5.2 Homological Algebra	PDF
homotopy groups; Eilenberg–MacLane spaces; the Hurewicz theoremOr: Spheres in spaces; They are not spheres; Homotopy and homology	§5.3 Horizons	PDF 1,PDF 2,PDF 3
homology categorifyingOr: Modules, polynomials and numbers	§5.3 Horizons	PDF

The shape of a question, revisited

This book began, in §0.1 The Shape of a Question, with a lump of clay, a thumb pushed through it, and two questions: what does it mean for two shapes to be “the same”, and why would anyone count holes?

To the first question we now have a layered answer. Two shapes can be equal, homeomorphic, or homotopy equivalent — and each of these is “isomorphic” in a suitable category. Homology and cohomology are functors: they cannot see the difference between equivalent shapes, so whenever they report a difference, the difference is real. That is why a computation with matrices can prove that a coffee cup is not a sphere, that every map of a disk to itself has a fixed point, and that you cannot comb a hairy ball.

To the second question the answer has grown in the telling. A hole turned out to be a cycle that bounds nothing; counting holes became computing $ker \partial / im \partial;$ and the same pattern — something that is locally fine but globally obstructed — reappeared as an impossible staircase, a vortex, a twisted bundle, a sensor network with a gap, a ranking that cannot be made consistent, a knot that cannot be untied. Cohomology is the general measure of that gap, and counting holes turned out to be a way of understanding why things that work everywhere locally can fail to work everywhere at once.

Along the way the book kept restaging its four recurring ideas: quotients (identify things — cycles modulo boundaries), kernels and images (what is crushed, what is reached), formal sums (chains as inventories of pieces), and reversed arrows (preimages, transposes, pullbacks, cochains). If those four ideas now feel like old friends, then the book has done what Thurston asked of mathematics in the epigraph: helped you understand and think more clearly.

Which road you take from here is up to you; the study guide above sketches three. On any of them you will keep meeting the clay ring of the first page. The loop that a computer finds in a cloud of data is a cycle that bounds nothing. The plateaus of conductance that physicists measure in a cold, thin layer of semiconductor count Chern numbers: the global twist of a bundle that is untwisted in every small patch. The league that cannot be ranked has a cohomology class where its missing games should be. When Poincaré began counting holes in higher dimensions, in 1895, he had none of these in view. You can now open a book about any of them and recognise the language on its first page. It is the language of a thumb pushed through a lump of clay, made precise.

Hodge theory: the best representative

The smooth version

The discrete version is least squares

Ranking with cohomology: HodgeRank

Circular coordinates from data

Generalized cohomology theories

K-theory: counting vector bundles

Cobordism: homology made of manifolds

The unifying idea: representability and spectra

Homotopy groups: the harder cousin

The Hopf fibration

Sheaf cohomology and the rising sea

Homology at work

Explorations

Where to go next: a study guide

The pure path: topology, geometry, algebra

The applied path: data, networks, computation

The physics path: fields, phases, knots

Watch alongside: companion videos

The shape of a question, revisited

References