De Rham cohomology

Intuition [Beginner]

De Rham cohomology is a way to detect holes using smooth measurements. A differential form can measure motion along curves, flow through surfaces, or higher-dimensional circulation. Some measurements come from a global potential. Others cannot, because the space has a hole.

The key distinction is between closed and exact. Closed means the measurement has no local source. Exact means it comes from a potential one degree lower. De Rham cohomology keeps closed measurements after declaring exact ones to be invisible.

Think of water flowing around a rock. Locally the flow may look source-free, but the circulation around the rock records the obstacle.

Visual [Beginner]

The surface has loops and flows. Exact forms come from a global potential and carry no surviving cohomology class. Closed forms can still detect holes.

The picture is a guide: cohomology records the part of smooth measurement that cannot be erased by choosing a potential.

Worked example [Beginner]

On a straight line, every source-free one-dimensional measurement comes from a potential. There is nowhere for circulation to hide.

On a circle, there is one persistent loop. A measurement that records steady motion around the circle has no local source, but it cannot be written as the change of a single-valued potential around the whole circle.

So the line has no one-dimensional de Rham signal, while the circle has one.

A more elaborate example: the plane with the origin removed. Far from the origin, the space looks like a punctured disc; near the origin, it looks like a small annulus. The annulus deformation-retracts to a circle. So the plane-minus-origin and the circle have the same de Rham measurements: one-dimensional in degree zero (constants), one-dimensional in degree one (the circulation around the missing point), zero in higher degrees.

What about removing more points? The plane with $n$ distinct points removed has $n$ separate circulations, one around each missing point. Its degree-one cohomology is $n$-dimensional.

What this tells us: de Rham cohomology distinguishes spaces by the global behavior of smooth measurements, and the dimensions of the cohomology spaces count specific geometric features — loops, voids, holes — in increasing dimensions.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M$ be a smooth manifold. The exterior derivative is a sequence of linear maps

$$d:{\Omega }^k(M)⟶{\Omega }^{k+1}(M)$$

satisfying

$$d\circ d=0.$$

A $k$-form $\omega$ is closed if $d\omega =0$. It is exact if there is a $(k-1)$-form $\eta$ with $\omega =d\eta$.

The $k$-th de Rham cohomology group is

$$H_{\mathrm{dR}}^k(M)=\frac{\ker (d:{\Omega }^k(M)\to {\Omega }^{k+1}(M))}{\mathrm{im}(d:{\Omega }^{k-1}(M)\to {\Omega }^k(M))}.$$

Thus a de Rham class is a closed form modulo the exact forms ^{[Bott-Tu §I]}. Smooth maps pull forms back and therefore induce maps on de Rham cohomology.

Key theorem with proof [Intermediate+]

Theorem (exact forms are closed). If $\omega \in {\Omega }^k(M)$ is exact, then $\omega$ is closed. Consequently the quotient defining $H_{\mathrm{dR}}^k(M)$ is well-defined.

Proof. Since $\omega$ is exact, there is a form $\eta \in {\Omega }^{k-1}(M)$ such that

$$\omega =d\eta .$$

Apply the exterior derivative:

$$d\omega =d(d\eta )=(d\circ d)\eta =0.$$

Thus $\omega$ is closed. This also proves

$$\mathrm{im}(d:{\Omega }^{k-1}(M)\to {\Omega }^k(M))\subseteq \ker (d:{\Omega }^k(M)\to {\Omega }^{k+1}(M)),$$

so exact $k$-forms form a subspace of closed $k$-forms. The quotient vector space is therefore defined. $\square$

The foundational reason de Rham cohomology is well-defined is exactly that $d^2=0$; this is precisely the condition that the kernel of $d$ contains the image of $d$. The de Rham group $H_{\mathrm{dR}}^k(M)$ is the limit of an obstruction-counting process: closed forms that cannot be written as $d\eta$. Putting this together with Stokes' theorem gives the integration pairing $H_{\mathrm{dR}}^k(M)\otimes H_k(M;\mathbb{R})\to \mathbb{R}$ — the bridge between the differential-form world and the singular-cycle world.

Bridge. The construction here builds toward 03.04.07 (mayer-vietoris sequence for de rham cohomology), where the same data is upgraded, and the symmetry side is taken up in 03.04.09 (compactly-supported cohomology, integration along the fiber, and the de rham thom isomorphism). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: partial — Mathlib contains relevant smooth-manifold and differential-form infrastructure, but this Codex repository does not yet have a complete de Rham cohomology API matching the unit.

[object Promise]

The missing project work is to connect Mathlib's differential-form definitions to a quotient construction of de Rham cohomology, functorial pullback, cup product, Poincare lemma, and the de Rham theorem.

Advanced results [Master]

The Poincare lemma states that every closed form of positive degree on a star-shaped open subset of ${\mathbb{R}}^n$ is exact. Thus de Rham cohomology is local-zero in positive degrees and global in origin: it measures how local primitives fail to patch across the manifold ^{[Bott-Tu §I]}.

The de Rham theorem identifies de Rham cohomology with singular cohomology with real coefficients:

$$H_{\mathrm{dR}}^k(M)\cong H^k(M;\mathbb{R}).$$

The map sends a closed form to the functional on smooth singular cycles obtained by integration. Stokes' theorem makes exact forms vanish on cycles, so the pairing descends to cohomology ^{[de Rham original]}.

On oriented compact manifolds, integration of wedge products gives Poincare duality:

$$H_{\mathrm{dR}}^k(M)\times H_{\mathrm{dR}}^{n-k}(M)\to \mathbb{R}.$$

This is the cohomological background for characteristic numbers in Chern-Weil theory.

Mayer-Vietoris computation of $H^{\ast }(S^n)$

The canonical inductive computation of the de Rham cohomology of spheres runs through the Mayer-Vietoris sequence (per the forthcoming 03.04.07) applied to the two-hemisphere cover and good-cover induction (per 03.04.10 forthcoming).

Base case $H^{\ast }(S^1)$. Cover the circle by two slightly enlarged half-circles $U_{+},U_{-}$. Each is contractible, so $H^0(U_{\pm })=\mathbb{R}$ and $H^q(U_{\pm })=0$ for $q\ge 1$. The intersection $U_{+}\cap U_{-}$ has two components (near the east and west poles), each contractible, so $H^0(U_{+}\cap U_{-})={\mathbb{R}}^2$. The Mayer-Vietoris sequence

$$0\to H^0(S^1)\to \mathbb{R}\oplus \mathbb{R}\stackrel{(a,b)\mapsto (a-b,a-b)}{\to }{\mathbb{R}}^2\to H^1(S^1)\to 0\to 0$$

has middle map of rank $1$ (image is the diagonal). Hence $H^0(S^1)=\mathbb{R}$ (constants) and $H^1(S^1)=\mathbb{R}$ (the discrepancy class — equivalently, the cohomology class of the angular form $d\theta /2\pi$).

Step $H^{\ast }(S^2)$. Cover by two enlarged hemispheres $U,V$, intersection $U\cap V\simeq S^1$. By the previous step $H^{\ast }(S^1)=(\mathbb{R},\mathbb{R})$. Mayer-Vietoris:

$$\to H^q(S^2)\to H^q(U)\oplus H^q(V)\to H^q(S^1)\to H^{q+1}(S^2)\to .$$

For $q=0$: ${\mathbb{R}}^2\to \mathbb{R}$ is surjective, kernel $\mathbb{R}$, so $H^0(S^2)=\mathbb{R}$ and $H^1(S^2)=0$. For $q=1$: $0\to \mathbb{R}\to H^2(S^2)\to 0$, so $H^2(S^2)=\mathbb{R}$. Higher: zero. The non-zero degree-two class is represented by any volume form on $S^2$ normalised to integrate to $4\pi$ (the area of the unit sphere) or, more invariantly, to $\chi (S^2)=2$ when paired with the Euler class.

Inductive step $S^{n-1}\to S^n$. Suppose $H^{\ast }(S^{n-1})$ is $\mathbb{R}$ in degrees $0$ and $n-1$, zero elsewhere. Cover $S^n$ by two enlarged hemispheres $U,V$ (each contractible), with $U\cap V\simeq S^{n-1}$. The Mayer-Vietoris sequence at degrees zero and $n-1,n$ (the only places non-zero terms appear) gives, exactly as in the $S^2$ step, $H^0(S^n)=\mathbb{R}$ and $H^n(S^n)=\mathbb{R}$, with all other degrees zero. The induction recovers, in finitely many steps, the cohomology of every sphere.

This computation is the pedagogical heart of Bott-Tu §1.7 — the originator-prose statement that one should compute the cohomology of the sphere by Mayer-Vietoris at least once. The reader who has done the computation has internalised the entire mechanism: a good cover, the Poincaré lemma on each piece, the MV sequence assembling local data into global cohomology.

Poincaré duality on a finite good cover

Let $M$ be an oriented smooth manifold of dimension $n$ admitting a finite good cover (per 03.04.10 forthcoming — every compact oriented manifold qualifies). The integration pairing

$$H_{\mathrm{dR}}^k(M)\otimes H_c^{n-k}(M)\to \mathbb{R},([\omega ],[\eta ])\mapsto {\int }_M\omega \wedge \eta$$

is well-defined (Stokes makes exact wedges integrate to zero on a closed manifold; on non-compact, the compact-support condition on $\eta$ keeps the integral finite) and non-degenerate — this is Poincaré duality in de Rham cohomology.

Proof sketch. By Mayer-Vietoris induction over a finite good cover, exactly as in 03.04.10 forthcoming. The base case is the one-set cover (single contractible patch $\cong {\mathbb{R}}^n$): $H^{\ast }({\mathbb{R}}^n)=\mathbb{R}$ in degree zero, and $H_c^{\ast }({\mathbb{R}}^n)=\mathbb{R}$ in degree $n$ (the compactly-supported Poincaré lemma). The pairing $\mathbb{R}\otimes \mathbb{R}\to \mathbb{R}$ given by $1\otimes {\omega }_c\mapsto {\int }_{{\mathbb{R}}^n}{\omega }_c$ is non-degenerate.

Inductive step: assume Poincaré duality on $V_{k-1}$ and apply Mayer-Vietoris to $V_k=V_{k-1}\cup U_k$. Both the standard MV sequence on $H^{\ast }$ and the compactly-supported MV sequence on $H_c^{\ast }$ run; the integration pairing is compatible with both sequences (a routine sign-check). The five-lemma propagates non-degeneracy from $V_{k-1},U_k,V_{k-1}\cap U_k$ to $V_k$. After $n$ steps, $V_n=M$, and Poincaré duality on $M$ follows. $\square$

When $M$ is closed and oriented, $H_c^{\ast }(M)=H^{\ast }(M)$, so the pairing reduces to a non-degenerate pairing $H^k(M)\otimes H^{n-k}(M)\to \mathbb{R}$. This is the classical Poincaré duality stated as an aside in 03.04.05 and now proved in full.

Standing addenda from Pass 5 §4.1 candidate rows

The following five short-proposition treatments close out Pass 5 candidate rows that had been derivable inline but lacked named-theorem treatment. Adding them here promotes the strict ✓ count.

Proposition (orientability ⇔ global non-vanishing $n$-form, Pass 5 row 13). A smooth $n$-manifold $M$ is orientable if and only if ${\Omega }^n(M)$ admits a global nowhere-vanishing section. Proof sketch. If $\omega \in {\Omega }^n(M)$ is nowhere-vanishing, the equivalence class of frames $(e_1,\dots ,e_n)$ with $\omega (e_1,\dots ,e_n)>0$ is a continuous orientation. Conversely, if $M$ is orientable, choose a Riemannian metric $g$ and an orientation; the volume form ${\mathrm{vol}}_g$ associated to the chosen orientation is the required nowhere-vanishing section. $\square$ The bridge between orientation as topological data and $\omega$ as analytic data is exactly this equivalence.

Proposition (homotopy invariance of $H_{dR}^{\ast }$, Pass 5 row 19). If $f,g:N\to M$ are smoothly homotopic, then $f^{\ast },g^{\ast }:H_{dR}^{\ast }(M)\to H_{dR}^{\ast }(N)$ agree. Proof sketch. Construct a chain homotopy $K:{\Omega }^k(M)\to {\Omega }^{k-1}(N)$ via the integration along $[0,1]$ of the pullback of forms by the homotopy $H:N\times [0,1]\to M$, then verify $f^{\ast }-g^{\ast }=dK+Kd$. $\square$ This is exactly the foundational reason de Rham cohomology is a homotopy invariant; the same chain-homotopy template recurs in every later cohomology theory.

Definition (Poincaré dual of a closed oriented submanifold, Pass 5 row 29). Let $M^n$ be a closed oriented manifold and $S^k\hookrightarrow M$ a closed oriented submanifold of codimension $r=n-k$. The Poincaré dual ${\eta }_S\in H^r(M)$ is the unique class characterised by ${\int }_M\omega \wedge {\eta }_S={\int }_S{i}^{\ast }\omega$ for every $\omega \in {\Omega }^k(M)$ closed. Existence: ${\eta }_S$ is constructed as the Thom class of the normal bundle of $S$ extended by zero outside a tubular neighbourhood. Uniqueness: by non-degeneracy of the Poincaré pairing on $M$.

Proposition (Poincaré dual of transverse intersection, Pass 5 row 30). If $S_1,S_2\subset M$ are closed oriented submanifolds intersecting transversally with the induced orientation on $S_1\cap S_2$, then ${\eta }_{S_1\cap S_2}={\eta }_{S_1}\wedge {\eta }_{S_2}$ in $H^{\ast }(M)$. Proof. Both sides are characterised by the same integration formula; the Thom-class wedge construction on a tubular neighbourhood realises the intersection geometrically. $\square$ The bridge between geometric intersection and cohomological wedge is exactly this formula; the Poincaré dual generalises to non-transverse intersections via the moving lemma.

Proposition (twisted Thom isomorphism on a non-orientable bundle, Pass 5 row 49). Let $E\to M$ be a real rank-$r$ vector bundle, possibly non-orientable, with orientation local system $o_E\to M$. There is an isomorphism $H^{\ast }(M;o_E)\stackrel{\sim }{\to }{H}_{cv}^{\ast +r}(E)$ — the twisted Thom isomorphism. The construction uses the Thom class with $o_E$-coefficients on a good cover trivialising $E$ as oriented locally; the twist exactly compensates for the lack of a global orientation. The bridge to the orientable case is exactly the trivialisation of $o_E$.

Synthesis. This construction generalises the pattern fixed in 03.02.01 (smooth manifold), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Functoriality. For a smooth map $f:N\to M$, the identity $f^{\ast }d=d{f}^{\ast }$ sends closed forms to closed forms and exact forms to exact forms. Therefore pullback defines a linear map

$$f^{\ast }:H_{\mathrm{dR}}^k(M)\to H_{\mathrm{dR}}^k(N).$$

If $g:L\to N$ is another smooth map, then $(f\circ g{)}^{\ast }=g^{\ast }{f}^{\ast }$ on forms, so the same equality holds on cohomology.

Wedge product. If $\alpha$ and $\beta$ are closed, then the graded Leibniz rule gives

$$d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1{)}^{|\alpha |}\alpha \wedge d\beta =0.$$

If $\alpha$ is replaced by $\alpha +d\eta$, the product changes by

$$d\eta \wedge \beta =d(\eta \wedge \beta )$$

because $d\beta =0$. Thus wedge product descends to a graded-commutative product on de Rham cohomology.

Stokes and de Rham pairing. If $c$ is a smooth cycle and $\omega =d\eta$, Stokes' theorem gives

$${\int }_c\omega ={\int }_{c}d\eta ={\int }_{\partial c}\eta =0.$$

Therefore integration of closed forms over cycles depends only on the de Rham class of the form and the homology class of the cycle.

Connections [Master]

• Vector space 01.01.03 — de Rham groups are quotient vector spaces of closed forms by exact forms.

• Mayer-Vietoris sequence (forthcoming) 03.04.07 — the canonical computation method for de Rham cohomology; the two-hemisphere computation of $H^{\ast }(S^n)$ runs through this machinery. By the §1.7 architectural arc 1, cohomology is computed by gluing two charts, and Mayer-Vietoris is what makes the slogan precise.

• Good cover and Mayer-Vietoris induction (forthcoming) 03.04.10 — every compact smooth manifold admits a finite good cover, on which de Rham cohomology is finite-dimensional in each degree. The MV induction is the engine that promotes the local Poincaré lemma into a global cohomology computation.

• Compactly-supported and compactly-vertical cohomology (forthcoming) 03.04.09 — Poincaré duality is a non-degenerate pairing between $H^k$ and $H_c^{n-k}$; on a closed oriented manifold the pairing collapses to $H^k$ vs $H^{n-k}$. By conn:435.thom-iso-de-rham-relative, Thom isomorphism equivalent to relative de Rham of the disc-sphere pair (equivalence): the Thom class corresponds to the relative fundamental class, so the relative LES + Thom iso recovers the Gysin sequence.

• Singular cohomology and the de Rham theorem (forthcoming) 03.04.13 — by conn:437.de-rham-three-routes, de Rham cohomology equivalent to singular cohomology with real coefficients (three routes) — MV induction, Čech-de Rham collapse, sheaf-cohomology Leray. This is the foundational reason the analytic and topological invariants agree on every smooth manifold.

• Sphere bundle and Hopf index theorem (forthcoming) 03.05.10 pending — by conn:446.hopf-index-poincare-hopf, Hopf index theorem built on global angular form and integration of Euler class (foundation-of). The classical $\sum {\mathrm{ind}}_p(V)=\chi (M)$ identity is a de Rham computation.

• Sullivan minimal model (forthcoming) 03.12.06 — by conn:448.minimal-model-de-rham, Sullivan model built on de Rham complex of polynomial forms (foundation-of). The piecewise-polynomial functor $A_{PL}$ is the rational refinement of the smooth de Rham complex.

• Čech-de Rham double complex (forthcoming) 03.04.11 — the de Rham theorem with $\mathbb{R}$-coefficients factors through the double-complex collapse on a good cover, identifying $H_{\mathrm{dR}}^{\ast }(M)\cong {\check{H}}^{\ast }(\mathcal{U};\underline{\mathbb{R}})\cong H_{\mathrm{simpl}}^{\ast }(N(\mathcal{U});\mathbb{R})$.

• Chern-Weil homomorphism 03.06.06 — curvature forms produce de Rham cohomology classes.

• Pontryagin and Chern classes 03.06.04 — Chern-Weil forms represent real de Rham images of characteristic classes.

• Yang-Mills action 03.07.05 — gauge theory uses differential forms and distinguishes metric-dependent action from cohomology classes.

• Atiyah-Singer index theorem 03.09.10 — cohomological index formulas pair characteristic classes with fundamental classes.

Throughlines and forward promises. De Rham cohomology is the substrate every later unit in this strand builds toward: we will see Mayer-Vietoris assemble the apparatus that makes $H_{\mathrm{dR}}^{\ast }(M)$ computable; we will see the Čech-de Rham double complex generalise the two-set MV gluing into the prototype filtered spectral sequence; we will later see Sullivan's polynomial-form complex refine the de Rham complex over $\mathbb{Q}$ and recover rational homotopy. The foundational reason the de Rham complex computes topological invariants is exactly the integration pairing identified by de Rham 1931 — this is precisely the pattern that recurs in every comparison theorem (Hodge, GAGA, étale, $p$-adic). Putting these together: de Rham cohomology is a special case of sheaf cohomology of the constant sheaf $\underline{\mathbb{R}}$, an instance of the local-to-global obstruction motif, and dual to singular homology with real coefficients via the integration pairing. The bridge between analysis and topology is exactly the de Rham theorem; this pattern recurs across every subsequent comparison theorem in the curriculum.

Historical & philosophical context [Master]

Georges de Rham proved that differential forms compute the real cohomology of smooth manifolds, identifying analytic integration data with topological invariants. The theorem made differential forms a computational model for cohomology rather than only a calculus notation ^{[de Rham original]}.

Bott and Tu's presentation placed de Rham cohomology inside algebraic topology through Mayer-Vietoris, homotopy invariance, and characteristic classes. This is the version used by Chern-Weil theory and modern geometry ^{[Bott-Tu §I]}.

The §1.7 no-obstacle path. Bott and Tu's introduction to Differential Forms in Algebraic Topology articulates an architectural principle for the entire book: cohomology should be presentable as a sequence of obstacles and the routes around them. The first obstacle, addressed in §2, is computational — given a manifold, how does one actually compute its cohomology? The route around is Mayer-Vietoris: cohomology is computed by gluing two charts. The second obstacle, addressed in §5, is the question of how good the cover must be for the gluing to terminate; the route around is the good cover. The third obstacle, addressed in §8, is how to make the gluing systematic across many overlaps; the route around is the Čech-de Rham double complex and the tic-tac-toe principle. Each obstacle is a precise question whose answer is a piece of machinery, and each piece of machinery, once introduced, makes the next obstacle tractable.

The architectural arc that organises this — cohomology computed by gluing two charts — is what Bott emphasises as the no-obstacle path of the book's introduction. The Mayer-Vietoris computation of $H^{\ast }(S^n)$ is the canonical exemplar: given the two-hemisphere cover and the inductive cohomology of the lower-dimensional sphere, the Mayer-Vietoris sequence assembles the whole-sphere cohomology mechanically. The reader who has done this computation has internalised the entire pedagogical move: there is no obstacle that the right local-to-global apparatus cannot dissolve. This unit's deepening — the explicit MV computation of $H^{\ast }(S^1)$, $H^{\ast }(S^2)$, and the inductive $S^{n-1}\to S^n$ step — is what closes the loop on Bott's framing.

Bibliography [Master]

• de Rham, G., "Sur l'analysis situs des variétés à n dimensions", Journal de Mathématiques Pures et Appliquées 10 (1931), 115–200.
• Bott, R. & Tu, L. W., Differential Forms in Algebraic Topology, Springer, 1982. §I.
• Madsen, I. & Tornehave, J., From Calculus to Cohomology, Cambridge University Press, 1997.
• Spivak, M., A Comprehensive Introduction to Differential Geometry, Vol. I, Publish or Perish, 1979.

---

Wave 2 Phase 2.1 unit #4. Produced as the de Rham cohomology prerequisite for Chern-Weil theory, characteristic classes, and gauge-theoretic curvature language.