Mayer-Vietoris sequence for de Rham cohomology

Intuition [Beginner]

Mayer-Vietoris is the rule that says: if you split a space into two overlapping pieces, the cohomology of the whole space can be assembled from the cohomology of each piece and of the overlap. It is the local-to-global gluing law for de Rham cohomology.

Cover a smooth space with two open sets $U$ and $V$. Each piece has its own cohomology — the patterns of closed measurements that survive on it. So does the overlap $U\cap V$. The Mayer-Vietoris rule weaves these three pieces of data into a single chain of vector spaces, with arrows running back and forth, telling you exactly which classes on the whole space project to which classes on the pieces, and which classes on the overlap fail to extend back to the whole.

Think of a circle covered by a slightly long northern arc and a slightly long southern arc, overlapping in two short bands at east and west. Each arc by itself is a stretched-out line — it has only a single constant cohomology class. But the two-piece overlap has two components, each contributing its own constant. The arithmetic of how the two pieces' constants compare on the two overlap components is exactly what produces the one extra class of the whole circle — its single non-zero one-form class. The rule that makes this arithmetic precise is Mayer-Vietoris.

Visual [Beginner]

A surface split into two overlapping caps. Their overlap is shown as a thin strip. Three boxes appear, one for each piece (left cap, right cap, overlap), connected by arrows that loop back through a fourth box for the whole surface.

The picture is a pipeline: cohomology classes flow in, get split into pieces, get compared on the overlap, and the discrepancies feed back into a higher-degree class on the whole.

Worked example [Beginner]

Take the circle and cover it with two slightly-extended half-circles that overlap in two short arcs near the east and west poles. Each half-circle is contractible, so its zero-degree de Rham measurements are one-dimensional (the constants) and its one-degree measurements vanish. The overlap has two components, each contractible, so its zero-degree measurements are two-dimensional (one constant per component) and its one-degree measurements vanish.

Now plug into the Mayer-Vietoris chain. At degree zero: a constant on the whole circle restricts to the same constant on each half-circle, and the two constants must agree on each of the two overlap components. So the zero-degree pull-back is one-dimensional, matching the one-dimensional constant on the whole circle.

At the overlap-zero step: the overlap has two-dimensional zero-degree measurements (a pair of constants, one per component), but only one-dimensional worth of these come from comparing the two half-circle constants. The remaining one-dimensional discrepancy — the difference of the constants on the two overlap components — has nowhere to go in degree zero; it must feed into a degree-one class on the whole circle. That extra dimension is the famous one-dimensional first cohomology of the circle.

What this tells us: the circle has one non-zero degree-one cohomology class, and the Mayer-Vietoris rule produces it explicitly from the discrepancy on the two-component overlap.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M$ be a smooth manifold and let $U,V$ be open subsets with $M=U\cup V$. The Mayer-Vietoris short exact sequence of de Rham complexes is

$$0\to {\Omega }^{\ast }(M)\stackrel{({\iota }_U^{\ast },{\iota }_V^{\ast })}{\to }{\Omega }^{\ast }(U)\oplus {\Omega }^{\ast }(V)\stackrel{{ȷ}_U^{\ast }-{ȷ}_V^{\ast }}{\to }{\Omega }^{\ast }(U\cap V)\to 0,$$

where ${\iota }_U:U\hookrightarrow M$, ${\iota }_V:V\hookrightarrow M$, ${ȷ}_U:U\cap V\hookrightarrow U$, ${ȷ}_V:U\cap V\hookrightarrow V$ are the inclusions, and the maps act by pullback (restriction) ^{[Bott-Tu §2]}.

The first map sends a global form $\omega$ to the pair of its restrictions $(\omega {|}_U,\omega {|}_V)$. The second map sends a pair $(\alpha ,\beta )\in {\Omega }^{\ast }(U)\oplus {\Omega }^{\ast }(V)$ to the difference of restrictions $\alpha {|}_{U\cap V}-\beta {|}_{U\cap V}$.

Long exact Mayer-Vietoris sequence in de Rham cohomology. Applying the snake lemma to the short exact sequence of complexes, we get the long exact sequence

$$\cdots \to H_{\mathrm{dR}}^q(M)\to H_{\mathrm{dR}}^q(U)\oplus H_{\mathrm{dR}}^q(V)\to H_{\mathrm{dR}}^q(U\cap V)\stackrel{\delta }{\to }{H}_{\mathrm{dR}}^{q+1}(M)\to \cdots ,$$

where the connecting homomorphism $\delta$ is constructed as follows. Given $[\omega ]\in H_{\mathrm{dR}}^q(U\cap V)$, choose a partition of unity $\{{\rho }_U,{\rho }_V\}$ subordinate to the cover $\{U,V\}$. Set $\alpha ={\rho }_V\omega$ on $U$ (extended by zero outside $U\cap \mathrm{supp}{\rho }_V$, which makes it smooth on all of $U$) and $\beta =-{\rho }_U\omega$ on $V$. Then $\alpha -\beta =({\rho }_V+{\rho }_U)\omega =\omega$ on $U\cap V$, so $(\alpha ,\beta )$ is a pre-image of $\omega$ under the difference-of-restrictions map. The pair $(d\alpha ,d\beta )$ has zero difference on the overlap (since $d\omega =0$), so $d\alpha$ and $d\beta$ patch into a global $(q+1)$-form on $M$ — this global form is $\delta [\omega ]$.

Compactly-supported variant. For compactly supported forms, the arrows reverse direction:

$$0\to {\Omega }_c^{\ast }(U\cap V)\to {\Omega }_c^{\ast }(U)\oplus {\Omega }_c^{\ast }(V)\to {\Omega }_c^{\ast }(M)\to 0,$$

producing a long exact sequence

$$\cdots \to H_c^q(U\cap V)\to H_c^q(U)\oplus H_c^q(V)\to H_c^q(M)\stackrel{\delta }{\to }{H}_c^{q+1}(U\cap V)\to \cdots .$$

The same partition of unity supplies extension by zero in the appropriate direction (per 03.04.06).

Key theorem with proof [Intermediate+]

Theorem (Mayer-Vietoris exactness). Let $M$ be a smooth manifold and $\{U,V\}$ an open cover. The sequence

$$0\to {\Omega }^{\ast }(M)\stackrel{({\iota }_U^{\ast },{\iota }_V^{\ast })}{\to }{\Omega }^{\ast }(U)\oplus {\Omega }^{\ast }(V)\stackrel{{ȷ}_U^{\ast }-{ȷ}_V^{\ast }}{\to }{\Omega }^{\ast }(U\cap V)\to 0$$

is short exact.

Proof. Injectivity at $\Omega^(M)$.\ast If$\omega \in \Omega^*(M)$has$\omega|_U = 0$and$\omega|_V = 0$,then$\omega$iszerooneverypointof$M = U \cup V$,hence$\omega = 0$.

Exactness at $\Omega^(U) \oplus \Omega^(V)$. If $(\alpha ,\beta )$ comes from a global form $\omega$ via $\alpha =\omega {|}_U$, $\beta =\omega {|}_V$, then $\alpha {|}_{U\cap V}=\omega {|}_{U\cap V}=\beta {|}_{U\cap V}$, so the difference is zero. Conversely, suppose $\alpha {|}_{U\cap V}=\beta {|}_{U\cap V}$. Define $\omega$ on $M$ by $\omega =\alpha$ on $U$ and $\omega =\beta$ on $V$; the values agree on the overlap, and the resulting function is smooth because smoothness is a local property. Thus $\ker ({ȷ}_U^{\ast }-{ȷ}_V^{\ast })=\mathrm{im}({\iota }_U^{\ast },{\iota }_V^{\ast })$.

Surjectivity at $\Omega^(U \cap V)$.\ast Thisisthepartition-of-unityargument.Chooseapartitionofunity${\rho_U, \rho_V}$subordinateto${U, V}$.Given$\omega \in \Omega^*(U \cap V)$, define

$$\alpha ={\rho }_V\omega \in {\Omega }^{\ast }(U),\beta =-{\rho }_U\omega \in {\Omega }^{\ast }(V).$$

The form ${\rho }_V\omega$ is well-defined on $U$ because ${\rho }_V$ vanishes outside $\mathrm{supp}({\rho }_V)\subset V$, so ${\rho }_V\omega$ extends by zero from $U\cap V$ to $U$. The same applies to ${\rho }_U\omega$ on $V$. On the overlap,

$$\alpha -\beta =({\rho }_V+{\rho }_U)\omega =\omega ,$$

since ${\rho }_V+{\rho }_U=1$ on $M$. Hence $({ȷ}_U^{\ast }-{ȷ}_V^{\ast })(\alpha ,\beta )=\omega$. $\square$

The snake lemma applied to this short exact sequence of cochain complexes produces the long exact Mayer-Vietoris sequence in de Rham cohomology. The construction of the connecting homomorphism $\delta$ uses the partition of unity to choose a pre-image, then takes its differential — this is the standard snake-lemma zig-zag, executed concretely on differential forms.

Synthesis. The MV sequence is exactly the local-to-global obstruction motif made precise: a class on $U\cap V$ that fails to extend is precisely the obstruction the connecting map records. This is the core insight of de Rham-cohomological computation. The MV sequence generalises to arbitrary covers via the Čech-de Rham double complex.

Bridge. The construction here builds toward 03.04.10 (good covers, finite-dimensionality of de rham cohomology, and the mayer-vietoris induction), 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 has the long-exact-sequence machinery (HomologicalComplex.LongExactSequence) and partial differential-form support, but the de Rham specialisation of Mayer-Vietoris through partition of unity is not formalised. The Codex stub captures the API shape.

[object Promise]

The missing project work is to wire Mathlib's Mathlib.Topology.PartitionOfUnity to the Codex differential-form API and to instantiate the snake lemma on the resulting complex.

Advanced results [Master]

Generalised Mayer-Vietoris. The two-set MV sequence extends to a generalised MV sequence on an arbitrary open cover $\mathcal{U}=\{U_{\alpha }\}$, expressed as the row exactness of the Čech-de Rham double complex of the forthcoming 03.04.11. The connecting maps of the two-set version become the Čech differential $\delta$ in the generalised version. Both arise from the same partition-of-unity contracting homotopy.

Mayer-Vietoris induction over a finite good cover. A good cover (every finite intersection contractible) on a compact manifold gives, by induction on the number of cover sets, finite-dimensionality of $H_{\mathrm{dR}}^{\ast }(M)$ in each degree and the canonical inductive computation of $H^{\ast }(S^n)$. This argument is the entire content of the forthcoming 03.04.10 and powers the dual proofs of Künneth and Poincaré duality.

Compactly-supported MV. The compactly-supported variant has reversed arrows because compactly-supported forms have covariant functoriality (extension by zero), not contravariant (restriction). The same partition of unity supplies the injectivity at ${\Omega }_c^{\ast }(M)$. The compactly-supported MV is the engine of Poincaré duality and the Thom-isomorphism approach to characteristic classes (per the forthcoming 03.04.09).

Sheaf-cohomology MV. Replacing ${\Omega }^{\ast }$ by the chain complex of any fine sheaf $\mathcal{F}$ — or by the Godement resolution of any sheaf — produces an MV sequence in sheaf cohomology, valid on a two-set open cover with no contractibility hypothesis. The link between the sheaf-cohomology MV and the de Rham MV is the de Rham theorem 03.04.13 forthcoming, three routes to which are MV induction, Čech-de Rham collapse, and sheaf-cohomology Leray.

Mayer's and Vietoris's original work. Walther Mayer in 1929 and Leopold Vietoris in 1930 both worked on simplicial complexes: $X$ a polyhedron given as the union of two subcomplexes $A,B$, with the corresponding simplicial chain complexes $C_{\ast }(A)+C_{\ast }(B)=C_{\ast }(A\cup B)$ and $C_{\ast }(A)\cap C_{\ast }(B)=C_{\ast }(A\cap B)$. The exact sequence

$$0\to C_{\ast }(A\cap B)\to C_{\ast }(A)\oplus C_{\ast }(B)\to C_{\ast }(A\cup B)\to 0$$

— with chains, not cochains — produces the simplicial Mayer-Vietoris sequence in homology. The de Rham version of Bott-Tu is the cochain-level upgrade with compactly-supported variant: the same five-term short exact sequence, dualised and applied to a smooth cover.

Full proof set [Master]

Connecting homomorphism — explicit construction. Given $[\omega ]\in H^q(U\cap V)$, produce $\delta [\omega ]\in H^{q+1}(M)$ as follows. Choose $\{{\rho }_U,{\rho }_V\}$ a partition of unity subordinate to $\{U,V\}$. The form ${\rho }_V\omega$, defined originally on $U\cap V$, extends by zero to a form on $U$ (because ${\rho }_V$ vanishes near $\partial (U\cap V)\cap U\subset U\setminus V$). Similarly $-{\rho }_U\omega$ extends to $V$. Now compute

$$d({\rho }_V\omega )=d{\rho }_V\wedge \omega +{\rho }_{V}d\omega =d{\rho }_V\wedge \omega \text{on }U\cap V,$$

using $d\omega =0$. Similarly $d(-{\rho }_U\omega )=-d{\rho }_U\wedge \omega$ on $U\cap V$. On the overlap

$$d({\rho }_V\omega )-d(-{\rho }_U\omega )=d{\rho }_V\wedge \omega +d{\rho }_U\wedge \omega =d({\rho }_V+{\rho }_U)\wedge \omega =0,$$

since ${\rho }_V+{\rho }_U=1$. Hence $d({\rho }_V\omega )$ on $U$ and $d(-{\rho }_U\omega )$ on $V$ patch into a global $(q+1)$-form on $M$, the representative of $\delta [\omega ]$.

Closedness. $d(d({\rho }_V\omega ))=0$ on $U$, hence the patched form is closed on $M$.

Independence of representative. If $\omega =d\eta$ on $U\cap V$, then by the same construction ${\rho }_V\omega ={\rho }_{V}d\eta =d({\rho }_V\eta )-d{\rho }_V\wedge \eta$, so $d({\rho }_V\omega )=-d(d{\rho }_V\wedge \eta )$, exact on $U$. Hence $\delta (d\eta )=0$, so $\delta$ descends to cohomology.

Snake-lemma compatibility. The above construction is exactly the snake-lemma zig-zag: lift $\omega$ to $({\rho }_V\omega ,-{\rho }_U\omega )$, apply $d$, compute the resulting global $M$-class. Standard homological algebra confirms exactness of the resulting long sequence.

Naturality. For a smooth $f:M^{\prime }\to M$ pulled back to the cover $\{f^{-1}(U),f^{-1}(V)\}$, $f^{\ast }$ commutes with restriction, with the exterior derivative, and with multiplication by pulled-back partition functions $f^{\ast }{\rho }_U,f^{\ast }{\rho }_V$ (which form a partition of unity on $M^{\prime }$ subordinate to the pulled-back cover). Hence the diagram of MV sequences commutes.

Compactly-supported variant — proof. The sequence

$$0\to {\Omega }_c^{\ast }(U\cap V)\stackrel{(j_U,j_V)}{\to }{\Omega }_c^{\ast }(U)\oplus {\Omega }_c^{\ast }(V)\stackrel{i_U-i_V}{\to }{\Omega }_c^{\ast }(M)\to 0$$

is short exact, where $j_U,j_V$ are extension by zero (defined because $U\cap V$ has support compact in $U$ and in $V$) and $i_U,i_V$ are extension by zero (similarly to $M$). Surjectivity at ${\Omega }_c^{\ast }(M)$ is again the partition-of-unity argument: for $\omega \in {\Omega }_c^{\ast }(M)$, write $\omega ={\rho }_U\omega +{\rho }_V\omega$ with ${\rho }_U\omega \in {\Omega }_c^{\ast }(U)$ and ${\rho }_V\omega \in {\Omega }_c^{\ast }(V)$, then $i_U({\rho }_U\omega )-i_V(-{\rho }_V\omega )=\omega$. The connecting map similarly involves the partition.

Connections [Master]

• De Rham cohomology 03.04.06 — Mayer-Vietoris is the central computational tool for de Rham cohomology; without it, the cohomology of even the circle requires direct integration of every form. By the §1.7 architectural arc 1, cohomology is computed by gluing two charts — and Mayer-Vietoris is what makes this slogan precise.

• Good cover and Mayer-Vietoris induction (forthcoming) 03.04.10 — by conn:428.mv-sequence-good-cover, Mayer-Vietoris sequence is the foundation for good-cover induction (foundation-of). The MV sequence is the engine that good-cover induction iterates to produce finite-dimensionality of $H_{\mathrm{dR}}^{\ast }$ on every compact manifold; the good-cover induction is what extends the two-set version of MV to a global theorem.

• Čech-de Rham double complex (forthcoming) 03.04.11 — the row exactness of the double complex is the generalised MV sequence on an arbitrary cover; the connecting homomorphism of the two-set MV becomes the Čech differential $\delta$. The pattern recurs: every spectral sequence we will see has its origin in this two-set MV gluing.

• Singular cohomology and the de Rham theorem (forthcoming) 03.04.13 — by conn:429.mv-de-rham-singular, de Rham theorem built on Mayer-Vietoris induction over a good cover (foundation-of). The de Rham and singular MV sequences agree under the integration pairing, and the five lemma propagates the agreement on each cover element to the union — this is the foundational reason the integration map descends to a cohomology isomorphism.

• Stokes' theorem 03.04.05 — Stokes is the integration pairing that makes the MV connecting homomorphism well-defined: ${\int }_{\partial c}\omega ={\int }_{c}d\omega$ ensures that the de Rham class assigned to $\delta [\omega ]$ is independent of the partition-of-unity choice.

• Compactly-supported and compactly-vertical cohomology (forthcoming) 03.04.09 — the compactly-supported MV variant is what sets up the Poincaré-duality pairing and the Thom isomorphism; the same partition-of-unity machinery, applied to forms with compact support, runs the dual sequence.

The Mayer-Vietoris sequence is the canonical computation method for de Rham cohomology — the no-obstacle path of Bott-Tu §1.7. Cohomology is computed by gluing two charts; the obstruction to extending a class on the overlap to the whole space is exactly the connecting map. We will see this same obstruction-to-gluing motif recur in the Čech-de Rham double complex, where the two-set sequence becomes a row of an arbitrary-cover sequence; in the spectral sequence of a filtered complex, where the connecting map becomes a higher differential; and in the family-index theorem for fiber bundles, where the connecting map becomes the Gysin sequence.

Throughlines and forward promises. Mayer-Vietoris is the foundational engine of de Rham cohomology computation. We will see the two-set MV sequence iterated by good-cover induction in 03.04.10; we will see the row exactness of the Čech-de Rham double complex in 03.04.11 generalise this to arbitrary covers; we will later see the Gysin sequence emerge as the Serre-spectral-sequence shadow of MV on a sphere bundle. This is exactly the no-obstacle path of Bott-Tu §1.7. The foundational reason cohomology is computable by gluing two charts is precisely the partition-of-unity argument; this is the same pattern that recurs in singular MV, in compactly-supported MV, and in the abstract spectral-sequence machinery of 03.13.01. Putting these together: every theorem in the de Rham strand is a special case of MV-induction or a generalisation of it. The bridge between local Poincaré-lemma vanishing and global cohomological structure is exactly the MV connecting homomorphism — the foundational insight Mayer 1929 and Vietoris 1930 codified, and Bott-Tu 1982 made the central pedagogical move.

Historical & philosophical context [Master]

Walther Mayer's 1929 Über abstrakte Topologie (Monatshefte für Mathematik und Physik 36) and Leopold Vietoris's 1930 Über die Homologiegruppen der Vereinigung zweier Komplexe (Monatshefte 37) introduced the simplicial-homology version of the sequence that now bears their names. Mayer worked at the level of abstract chain complexes; Vietoris, motivated by the question of how to build the homology of a polyhedron from the homology of two subpolyhedra, gave the explicit short exact sequence

$$0\to C_{\ast }(A\cap B)\to C_{\ast }(A)\oplus C_{\ast }(B)\to C_{\ast }(A\cup B)\to 0$$

and derived the long exact sequence by hand. Both papers preceded the formal language of cochain complexes and snake lemmas; the proofs were direct combinatorial arguments on simplicial chains.

The de Rham version — for differential forms instead of simplicial chains, and on smooth manifolds rather than polyhedra — is implicit in de Rham's 1931 Sur l'analysis situs des variétés à $n$ dimensions, which used a partition-of-unity argument to prove that the de Rham cohomology of a smooth manifold equals the singular cohomology with real coefficients. The explicit cochain formulation of the Mayer-Vietoris sequence in the de Rham complex is the form pioneered in the 1950s by André Weil (in his Sur les théorèmes de de Rham, 1952) and made canonical by Bott and Tu in their 1982 textbook.

In Bott-Tu §2 the partition-of-unity argument receives its modern, definitive treatment. Bott's framing is what we channel here: cohomology is computed by gluing two charts — the first of the four §1.7 architectural arcs of Differential Forms in Algebraic Topology. The remarkable fact is that this single statement, executed via the two-set MV sequence, is the engine for everything else in the book: the inductive computation of $H^{\ast }(S^n)$, the finite-dimensionality of cohomology on a compact manifold, the Künneth formula, the Poincaré-duality pairing, the Thom isomorphism, and ultimately the spectral-sequence machinery of Chapter III. The sequence is the no-obstacle path: every later result either applies the sequence directly or generalises it (to arbitrary covers, to compactly-supported forms, to filtered complexes).

The pedagogical point Bott emphasises is that the MV sequence is so simple — three vector spaces with arrows between them — that the reader can compute the cohomology of every standard example by direct application. The reader has, with no machinery, computed $H^{\ast }(S^1),H^{\ast }(S^2)$, and $H^{\ast }$ of any space built up from contractible pieces by induction. This is what makes the de Rham complex a computational tool, not merely a definition. Without Mayer-Vietoris, the de Rham complex is a formal object; with it, the de Rham cohomology of any reasonable manifold becomes calculable by hand.

Bibliography [Master]

• Mayer, W., "Über abstrakte Topologie", Monatshefte für Mathematik und Physik 36 (1929), 1–42.
• Vietoris, L., "Über die Homologiegruppen der Vereinigung zweier Komplexe", Monatshefte für Mathematik und Physik 37 (1930), 159–162.
• de Rham, G., "Sur l'analysis situs des variétés à $n$ dimensions", Journal de Mathématiques Pures et Appliquées 10 (1931), 115–200.
• Weil, A., "Sur les théorèmes de de Rham", Commentarii Mathematici Helvetici 26 (1952), 119–145.
• Bott, R. & Tu, L. W., Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer-Verlag, 1982. §2 (Mayer-Vietoris exact sequence) and §5 (good covers + Mayer-Vietoris induction).
• Hatcher, A., Algebraic Topology, Cambridge University Press, 2002. §2.2 (singular-homology Mayer-Vietoris).
• Madsen, I. & Tornehave, J., From Calculus to Cohomology, Cambridge University Press, 1997. §5 (Mayer-Vietoris exact sequence).
• Eilenberg, S. & Steenrod, N., Foundations of Algebraic Topology, Princeton University Press, 1952 (axiomatic Mayer-Vietoris).

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Bott-Tu Pass 4 — Agent A — N1. Mayer-Vietoris exact sequence for de Rham cohomology, partition-of-unity proof of exactness, snake-lemma connecting homomorphism, compactly-supported variant, naturality. Originator-prose for Bott-Tu §2 + §5. Foundation for 03.04.10 (good-cover induction), 03.04.11 (Čech-de Rham double complex), 03.04.09 (Thom isomorphism via $H^_{cv}$), 03.04.12 (Künneth dual proof), 03.04.13 (de Rham theorem).*