de Rham Cohomology

The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology.

At the end of the last chapter you met a strange 1-form on the plane with one point removed. Its exterior derivative is zero everywhere — no swirl at any point — and yet its integral around a circle is $2\pi ,$ not zero. That is Escher’s staircase from §4.1 Cochains: Measuring Shapes all over again: honest at every step, impossible once round. This time it is built from calculus instead of numbers on edges, and the missing point is to blame.

This chapter turns that observation into an instrument for detecting holes with derivatives and integrals, called de Rham cohomology. Then it asks a question that has no right to a simple answer. You now own two hole detectors. One is made of smooth functions and limits; the other of finite lists of numbers written on the triangles of a triangulation. Do they ever disagree?

Closed and exact forms

A gradient field never swirls: if $\omega =df$ then $d\omega =d(df)=0,$ as §4.3 Differential Forms showed. Physics leans hard on the converse — a force with no curl, the textbooks say, comes from a potential energy — and on most regions the converse is true. This chapter is about the regions where it is false, and why. First, names for the two conditions.

For a 1-form on the plane, $\omega =Pdx+Qdy,$ being closed means $\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}:$ the curl of the field $(P,Q)$ vanishes everywhere. Being exact means $\omega =df$ for some function $f:$ the field is a gradient, and $f$ is its potential (the “height” of §4.1 Cochains: Measuring Shapes). Because $d(df)=0,$ every exact form is closed. Everything below hangs on the converse.

The loop test

There is a clean way to tell exactness from closedness, using loop integrals. If $\omega =df$ is exact, then for every closed loop $\gamma ,$

Conversely, suppose every loop integral of $\omega$ vanishes, on a connected region. Fix a base point $p_0$ and define $f(p)={\int }_{p_0}^p\omega ,$ integrating along any path from $p_0$ to $p.$ Two different paths give the same answer, because going out along one and back along the other is a loop, whose integral is zero. And $df=\omega :$ near $p,$ $f$ changes by the integral of $\omega$ along short steps, which is what $df=\omega$ says. So:

Closedness is a local condition — you can check it at each point by computing derivatives. Exactness is a global condition — it asks for a single function on the whole region. Figure 4.4.1 lets you test both on five forms; drag the loop around and watch the two numbers.

This should feel familiar. In §4.1 Cochains: Measuring Shapes you met edge labellings with zero “curl” around every filled triangle which were nevertheless not differences of vertex heights, because a loop around an unfilled hole picked up a nonzero sum. Here is the dictionary between the two worlds:

The de Rham cohomology groups

Following the recipe of Part III and §4.2 Cohomology Groups, we measure the gap between closed and exact forms with a quotient. The closed $k$-forms $Z^k(M)=\ker (d:{\Omega }^k(M)\to {\Omega }^{k+1}(M))$ form a vector space, the exact ones $B^k(M)=\mathrm{im}(d:{\Omega }^{k-1}(M)\to {\Omega }^k(M))$ form a subspace of it, and we divide.

In words: take all the closed forms, and agree to ignore the ones that are exact, since they are “closed for a trivial reason”. Whatever is left over measures closed forms that fail to be exact for a global reason. Concretely, to show that two closed forms $\omega$ and ${\omega }^{\prime }$ represent the same class you exhibit a form $\eta$ with $\omega -{\omega }^{\prime }=d\eta ;$ to show that they do not, you usually find a cycle over which they integrate differently — since, by Stokes’ theorem, an exact form integrates to zero over every cycle. This is the same quotient construction as $\mathbb{Z}/n$ in §1.2 Equivalence and Quotients and $V/W$ in §1.5 Linear Algebra: declare two things the same when they differ by something negligible. Over the real numbers these groups are vector spaces, and the most useful thing to know about each is its dimension.

Degree zero: what does not change

A 0-form is a function $f,$ and there are no forms of degree $-1,$ so nothing is exact in degree $0.$ A function is closed when $df=0,$ that is, when all its partial derivatives vanish: it is constant near every point. Such locally constant functions are constant on each connected piece of $M$ but may take different values on different pieces. So, when $M$ has finitely many components,

one real number for each piece, exactly as $H^0$ of a graph counted its components in §4.1 Cochains: Measuring Shapes. At the other end, if $M$ has dimension $n$ there are no nonzero forms of degree $k>n,$ so $H_{\mathrm{dR}}^k(M)=0$ for $k>n.$

The line and the circle

The circle is more interesting. Describe a point of the circle $S^1$ by its angle $\theta ,$ remembering that $\theta$ and $\theta +2\pi$ are the same point. The angle itself is not a function on the circle — it has no single value at the starting point — but its differential is a perfectly good 1-form: it measures the angle swept by a small step, and that does not care which value of $\theta$ you started from. This form is written $d\theta ,$ and every 1-form on the circle is $g(\theta )d\theta$ for a smooth $2\pi$-periodic function $g.$

Compare this with §3.3 Homology Groups: the circle has $H_0\cong \mathbb{Z}$ and $H_1\cong \mathbb{Z}.$ De Rham cohomology finds the same component and the same hole, with $\mathbb{R}$ in place of $\mathbb{Z}.$

The vortex: a closed form that is not exact

Now the hero of the chapter. On the punctured plane ${\mathbb{R}}^2\setminus 0$ (the plane with the origin removed) consider

Away from the origin, the polar angle $\theta$ of a point can be chosen to vary smoothly, at least on any region that does not go all the way around. On the right half-plane $x>0,$ for example, $\theta =\arctan (y/x).$ Let us check that its differential is $\omega .$ The derivative of $\arctan u$ is $1/(1+u^2),$ so by the chain rule

and these are exactly the coefficients of $\omega .$ On other half-planes a different formula for the angle does the same job (on the upper half-plane, $\theta =\mathrm{arccot}(x/y),$ say), and two such formulas differ by a constant where both are defined, so they have the same differential. That is why this form is called the angle form and is written $d\theta .$ Being locally the differential of a function, it is closed: $d\omega =d(d\theta )=0.$ (You can also check ${\partial }_{x}Q={\partial }_{y}P$ directly; it was the last exercise of the previous chapter.) Its field $(P,Q)$ is the vortex of Figure 4.3.2: it circles the origin with strength $1/r,$ and has no curl anywhere.

And yet around the unit circle, $x=\cos t,$ $y=\sin t,$ we have $\omega =({\sin }^{2}t+{\cos }^{2}t)dt=dt,$ so

By the loop test, $d\theta$ is not exact on the punctured plane. There is no function $f$ on ${\mathbb{R}}^2\setminus 0$ with $df=d\theta .$ The notation $d\theta$ is a trap: it is the derivative of the angle only locally, never globally.

The endless staircase

Imagine walking once around the origin while keeping track of your angle continuously. You start at $\theta =0;$ a quarter of the way round you are at $\pi /2;$ halfway at $\pi ;$ and when you are back where you started, your angle reads $2\pi ,$ not $0.$ Walk round again and it reads $4\pi .$ The angle is a multivalued function: a height you can climb for ever while going in circles. It is the staircase of Escher’s monks, which Lionel and Roger Penrose first drew in 1958 (Penrose & Penrose 1958). Theirs is an optical illusion, a staircase that pretends to close up. The angle is the real thing.

Figure 4.4.2 builds it honestly, in three dimensions. Over each point of the punctured plane, stack all its possible angles $\theta ,\theta \pm 2\pi ,\theta \pm 4\pi ,\dots$ as heights. These stacks form a single smooth spiral surface, the helicoid, like the ramp of a multistorey car park wound around the missing axis. It never pretends to close up; it keeps rising. A loop in the plane, together with a continuously chosen angle along it, traces a path on the ramp. If the loop goes around the hole, the path climbs a full storey per lap; if it does not, the path rises and falls and comes back to its start.

Winding numbers

The staircase also tells us what $\oint d\theta$ is around any loop. In §2.3 Homotopy you met the winding number $w(\gamma ,0)$ of a loop $\gamma$ around the origin: the net number of times $\gamma$ goes around, counting counterclockwise turns positively and clockwise ones negatively.

Integration has turned into counting. An integral is the kind of number you expect to drift when you nudge the loop, yet this one sits on a whole number of turns and refuses to budge. Remember that: it is the first hint of the surprise at the end of the chapter.

The first cohomology of the punctured plane

The same argument works with several holes. On the plane with $k$ points removed, each puncture has its own angle form, and $H_{\mathrm{dR}}^1\cong {\mathbb{R}}^k:$ a closed 1-form is determined, up to exact forms, by its $k$ integrals around small circles about the punctures. The “two vortices” form of Figure 4.4.1 is $d{\theta }_a-d{\theta }_b,$ with integrals $2\pi$ around $a,$ $-2\pi$ around $b,$ and $0$ around both.

The Poincaré lemma: no holes, no cohomology

If holes are what stop closed forms from being exact, then on a region without holes every closed form should be exact. The cleanest version of this uses a particularly simple kind of hole-free region.

For 1-forms the proof is a construction you can watch. Put the centre at the origin. To find a potential $f$ at a point $p,$ integrate $\omega$ along the straight ray from the centre to $p$ — the ray stays inside $U,$ so this makes sense:

The ray construction also shows why the annulus is different. From any centre $c$ in the annulus, some straight rays run into the hole, so the construction cannot reach the points behind it. Going around the hole instead, the potential would have to take two different values at the same point, one for each way around, and they differ by $\oint d\theta =2\pi .$

Invariance and cutting into pieces

Maps act backwards on cohomology

A smooth map $f:M\to N$ pulls forms back from $N$ to $M.$ Since pullback commutes with $d,$ it sends closed forms to closed forms ($d{f}^{\ast }\omega =f^{\ast }d\omega =0)$ and exact forms to exact forms ($f^{\ast }d\eta =d{f}^{\ast }\eta ),$ so it gives a well-defined linear map

The arrow points backwards — the fourth recurring idea once more — just as $f^{\ast }$ on cochains did in §4.2 Cohomology Groups. And composition behaves: $(g\circ f{)}^{\ast }=f^{\ast }\circ g^{\ast }.$

The idea is the one we used for loops above: a homotopy sweeps out a “cylinder” between $f$ and $g,$ and Stokes’ theorem on that cylinder shows that, for a closed form, what you see at one end differs from what you see at the other only by an exact form (Lee 2013, ch. 17). Here is what it buys immediately:

• ${\mathbb{R}}^n$ is homotopy equivalent to a point, so $H_{\mathrm{dR}}^k({\mathbb{R}}^n)=0$ for $k\ge 1$ — the Poincaré lemma again.
• The punctured plane deformation-retracts onto the unit circle, so $H_{\mathrm{dR}}^1({\mathbb{R}}^2\setminus 0)\cong H_{\mathrm{dR}}^1(S^1)\cong \mathbb{R},$ matching our computation.
• Space minus a straight line also retracts onto a circle, so $H_{\mathrm{dR}}^1({\mathbb{R}}^3\setminus \text{line})\cong \mathbb{R}.$ Space minus a point retracts onto a sphere, so $H_{\mathrm{dR}}^1({\mathbb{R}}^3\setminus 0)=0,$ but, as we will see, $H_{\mathrm{dR}}^2({\mathbb{R}}^3\setminus 0)\cong \mathbb{R}.$

Mayer–Vietoris: computing by cutting

Like homology in §3.6 Exact Sequences and Mayer–Vietoris, de Rham cohomology can be computed by cutting a space into overlapping pieces. If $M=U\cup V$ with $U,$ $V$ open, there is a long exact sequence (Bott & Tu 1982, §2)

(all groups de Rham). The first map restricts a form to both pieces; the second takes two forms on $U$ and $V$ and subtracts their restrictions to the overlap; the third, the “connecting map”, is the clever one.

Let us run it for the circle. The arcs $U$ and $V$ are each connected and contractible, so $H^0(U)=H^0(V)=\mathbb{R}$ and $H^1(U)=H^1(V)=0.$ Their intersection is two disjoint arcs $W_1\bigsqcup W_2,$ so $H^0(U\cap V)={\mathbb{R}}^2.$ A pair of constants $(a,b)$ on $U$ and $V$ is sent to the difference $b-a$ on each of $W_1$ and $W_2:$ the image of this map is only the diagonal line $\{(t,t)\}$ in ${\mathbb{R}}^2.$ Exactness at $H^0(U\cap V)$ says that the connecting map kills exactly this diagonal, so it carries the quotient ${\mathbb{R}}^2/\{(t,t)\}\cong \mathbb{R}$ injectively into $H^1(S^1).$ Exactness at $H^1(S^1),$ where the next group $H^1(U)\oplus H^1(V)$ is $0,$ says the connecting map is also onto. So $\dim H_{\mathrm{dR}}^1(S^1)=2-1=1.$

Geometrically, the leftover class is a choice of two constants $(c_1,c_2)$ on the two overlaps that is not of the form $(t,t):$ a “jump” across one overlap that is not matched by the same jump across the other. Smoothing such a jump out with a bump function produces a closed 1-form whose integral around the circle is the difference $c_2-c_1$ (up to a sign fixed by conventions) — a copy of $d\theta ,$ up to a constant multiple and exact forms.

De Rham’s theorem: calculus meets combinatorics

Step back and look at what you own. On one side are the de Rham groups: closed forms modulo exact ones, made of smooth functions, derivatives and limits. On the other are the groups $H^k(M;\mathbb{R})$ of §4.2 Cohomology Groups: cocycles modulo coboundaries, made from a triangulation (or from singular simplices) and nothing but lists of numbers, one per simplex.

Try both on the annulus around the missing origin. Its closed 1-forms fill an infinite-dimensional space — any multiple of $d\theta$ plus the differential of any smooth function you care to write down — and its exact forms fill another. Divide one by the other and, by homotopy invariance and our computation for the punctured plane, a single dimension survives, spanned by $[d\theta ].$ Now triangulate the same annulus with six vertices, twelve edges and six triangles, as in Figure 4.4.6. A 1-cochain is twelve numbers. The six triangle rules cut the cocycles down to a six-dimensional space; the coboundaries, height differences of six vertices, take up five of those dimensions. One dimension survives. Infinitely many smooth functions on one side, twelve numbers on the other, and the same answer. Coincidence? Integration says no.

Given a $k$-form $\omega ,$ integrate it over each oriented $k$-simplex $\sigma$ of a triangulation to get a number. That assignment is a $k$-cochain, $I(\omega )(\sigma )={\int }_{\sigma }\omega :$ the integration map, or de Rham map. Stokes’ theorem says

that is, $I\circ d=\delta \circ I.$ So integration sends closed forms to cocycles (if $d\omega =0$ then $\delta I(\omega )=I(d\omega )=0)$ and exact forms to coboundaries ($I(d\eta )=\delta I(\eta )),$ and therefore gives a linear map $H_{\mathrm{dR}}^k(M)\to H^k(M;\mathbb{R}).$

Look at the numbers the integration map produced for $d\theta .$ Each inner or outer edge subtends $120^{\circ }$ at the missing point, so it carries a third of a turn: $+\frac{1}{3}$ when its arrow runs counterclockwise, $-\frac{1}{3}$ when it runs clockwise. Each edge from an inner vertex $a_i$ to an outer vertex subtends $60^{\circ }$ and carries $\pm \frac{1}{6}.$ Around every triangle the angles cancel, since a triangle that does not contain the origin subtends no net angle. Around either loop they add up to one turn. Any other cocycle whose loop sum is one turn — say, the “fence of ones” of §4.1 Cochains: Measuring Shapes, which is $1$ on the three edges $a_0\to a_1,$ $a_0\to b_1,$ $b_0\to b_1$ and $0$ everywhere else — differs from this one by a coboundary: they are two representatives of the same class. Of all those representatives, the one produced by integrating $d\theta$ is the most evenly spread: it has the smallest sum of squares. That is a first glimpse of the Hodge theory sketched in §5.3 Horizons.

Pause on how unlikely this is. The cochain side depends on a triangulation: refine it and every cochain group changes size, although the quotients do not (the invariance of §3.5 Maps, Invariance, and First Triumphs). The form side has never heard of triangles. One theory is made of limits, the other of arithmetic you could do with a pencil, and on every smooth manifold, in every degree, they give the same real cohomology. We will not prove it. The proofs compare the two theories piece by piece with a Mayer–Vietoris argument, using the Poincaré lemma for the pieces; Bott & Tu (1982, §8) give a particularly clean version. Here is what it buys:

• Two worlds agree. The smooth world of forms and derivatives and the combinatorial world of simplices and matrices measure the same holes. Any calculation can be done in whichever world is easier.
• Periods detect everything. A closed form is exact if and only if all its periods — its integrals over cycles — vanish. We saw this for loops; de Rham’s theorem says it in every degree.
• No torsion. Forms have real coefficients, so de Rham cohomology cannot see the torsion that $\mathbb{Z}$-coefficients detect. For the projective plane, $H_{\mathrm{dR}}^k({\mathbb{RP}}^2)=\mathbb{R},0,0,$ although $H^2({\mathbb{RP}}^2;\mathbb{Z})=\mathbb{Z}/2$ (§4.2 Cohomology Groups).

Raoul Bott, remembering de Rham, called the theorem “a sort of topological form of the particle-wave equivalence of quantum mechanics” (O’Connor & Robertson 2015). One way to read him: a chain is concentrated on a few simplices, like a particle; a form is spread over the whole manifold, like a wave; and integration lets each describe the other completely.

Holes you can feel: physics

Closed forms that are not exact are not only a mathematician’s curiosity. Physics is full of fields with no local “curl” whose loop integrals are nonetheless not zero, and every time, a hole in space is responsible.

Conservative forces

A force field $\mathbf{F}$ is conservative if the work it does around every closed path is zero; then it is the (negative) gradient of a potential energy, $\mathbf{F}=-\nabla U,$ and energy is conserved. In the language of forms, the work 1-form is exact. Gravity near the ground, $U=mgy,$ is the classic example. On a region without holes, the Poincaré lemma says that “no curl” already guarantees a potential. On a region with holes it does not.

The field around a wire

A steady current $I$ flowing up a long straight wire along the $z$-axis produces a magnetic field circling the wire,

where ${\mu }_0$ is a physical constant. As a 1-form, this is $\frac{{\mu }_{0}I}{2\pi }d\theta$ on space with the wire removed. Outside the wire it has no curl, so it is closed. But around any loop it gives

which is Ampère’s law. So $\mathbf{B}$ is closed but not exact: there is no “magnetic potential function” outside a wire, and the obstruction is a class in $H_{\mathrm{dR}}^1({\mathbb{R}}^3\setminus \text{wire})\cong \mathbb{R}.$ For a current in a closed loop of wire, the number of times becomes the linking number of the two loops from §3.7 Knots and Links: Ampère’s law measures it with a magnetic field.

The Aharonov–Bohm effect

Physicists often describe a magnetic field $\mathbf{B}$ through a vector potential $\mathbf{A}$ with $\mathbf{B}=\mathrm{curl}\mathbf{A};$ as forms, a 1-form $A$ with $B=dA.$ Take a long thin solenoid — a coil of wire — carrying a magnetic flux $\Phi$ inside it. Outside the solenoid the magnetic field is zero, so $dA=0$ there: $A$ is closed. But by Stokes’ theorem on a disk spanning a loop around the solenoid, $\oint A=\Phi \ne 0.$ Outside the solenoid, $A=\frac{\Phi }{2\pi }d\theta$ (up to an exact form): the angle form once more.

Classically this would not matter, since a charged particle feels only $\mathbf{B},$ and $\mathbf{B}=0$ wherever the particle goes. Quantum mechanically it matters. Yakir Aharonov and David Bohm pointed out in 1959 (Aharonov & Bohm 1959), as Werner Ehrenberg and Raymond Siday had a decade earlier (Ehrenberg & Siday 1949), that the wave of a charged particle travelling around the solenoid acquires an extra phase $\frac{q}{\hslash }\oint A=\frac{q}{\hslash }\Phi$ per turn, which shifts the interference pattern of electrons passing on either side. The most convincing confirmation came in 1986, when Akira Tonomura and colleagues sent electrons around a tiny ring magnet sealed inside a superconducting shield (Tonomura et al. 1986). The electrons never touch the magnetic field. What they measure is the period $\oint A$ of the closed form $A$ around the hole — its cohomology class, read through a loop.

Gauss’s law and a hole of dimension two

One dimension up, consider space with the origin removed, ${\mathbb{R}}^3\setminus 0,$ and the 2-form

It is closed ($d\sigma =0,$ a computation you can try), and on the unit sphere it restricts to the area form, so ${\iint }_{S^2}\sigma =4\pi .$ If $\sigma$ were exact, $\sigma =d\eta ,$ then Stokes’ theorem on the sphere — which has no boundary — would give ${\iint }_{S^2}\sigma ={\iint }_{\partial S^2}\eta =0.$ So $\sigma$ is closed but not exact, and in fact $H_{\mathrm{dR}}^2({\mathbb{R}}^3\setminus 0)\cong \mathbb{R},$ generated by $[\sigma ].$ The integral of $\sigma$ over a closed surface is $4\pi$ times the number of times the surface wraps around the origin, which is why $\sigma$ is called the solid-angle form.

The electric field of a point charge $Q$ is exactly $\frac{Q}{4\pi {\epsilon }_0}\sigma ,$ and the statement “the flux out of a closed surface equals the enclosed charge over ${\epsilon }_0”$ is Gauss’s law. The divergence theorem of the previous chapter would give zero flux for every surface — if the field were defined everywhere inside. The charge is the hole.

Exercises

Summary

Further reading

• Differential Forms in Algebraic Topology — Raoul Bott and Loring Tu

  The classic: de Rham cohomology, Mayer–Vietoris, the Poincaré lemma, and how far forms can take you in topology. Graduate level, but its first chapter reads well after this one.
• An Introduction to Manifolds — Loring W. Tu

  The gentlest rigorous route: its closing chapters build de Rham cohomology, the Mayer–Vietoris sequence and homotopy invariance from scratch, with many worked computations. Written as a prequel to Bott and Tu.
• The Essence of de Rham Cohomology — Alice Petrov free

  A student-level exposition covering Mayer–Vietoris, homotopy invariance, Poincaré duality and de Rham’s theorem.
• Discrete Differential Geometry: An Applied Introduction — Keenan Crane free

  Shows forms becoming cochains on meshes — de Rham’s map made concrete — with the Hodge decomposition as a payoff.
• A Visual Introduction to Differential Forms and Calculus on Manifolds — Jon Pierre Fortney

  Its chapters on the Poincaré lemma and on electromagnetism treat closed and exact forms with many pictures.
• A Geometric Approach to Differential Forms — David Bachman free

  Gentle and geometric; includes a short section on de Rham cohomology as an application.
• Algebraic Topology — Allen Hatcher free

  For the combinatorial side of de Rham’s theorem: simplicial and singular cohomology with any coefficients.
• A path-dependent vector field with zero curl — Duane Nykamp (Math Insight) free

  A short interactive page on the vortex field, for readers who prefer the vector-calculus language.
• Significance of Electromagnetic Potentials in the Quantum Theory — Yakir Aharonov and David Bohm

  The 1959 paper predicting the Aharonov–Bohm effect (Physical Review 115, 485–491).
• The Feynman Lectures on Physics, Vol. II, ch. 15: The Vector Potential — Richard Feynman, Robert Leighton and Matthew Sands free

  Feynman’s own account of why the vector potential is real, including the solenoid experiment.