Proof of the Mayer-Vietoris Exact Sequence

The Mayer-Vietoris sequence is the primary computational tool in de Rham cohomology. Its proof is a careful application of the short exact sequence of differential form complexes together with the snake lemma.

Statement

Theorem5.2Mayer-Vietoris sequence

Let $M = U \cup V$ with $U, V$ open. There is a long exact sequence

\cdots \to H^k(M) \xrightarrow{r} H^k(U) \oplus H^k(V) \xrightarrow{s} H^k(U \cap V) \xrightarrow{\delta} H^{k+1}(M) \to \cdots

Proof

Step 1: Short exact sequence of complexes. Define the sequence of cochain complexes

0 \to \Omega^k(M) \xrightarrow{r} \Omega^k(U) \oplus \Omega^k(V) \xrightarrow{s} \Omega^k(U \cap V) \to 0,

where $r(\omega) = (\omega|_U, \omega|_V)$ and $s(\alpha, \beta) = \alpha|_{U \cap V} - \beta|_{U \cap V}$ .

Injectivity of $r$ : If $\omega|_U = 0$ and $\omega|_V = 0$ , then $\omega = 0$ on $U \cup V = M$ .

$\ker s = \operatorname{im} r$ : $s(\alpha, \beta) = 0$ means $\alpha|_{U \cap V} = \beta|_{U \cap V}$ , so $\alpha$ and $\beta$ patch to a global form $\omega$ with $r(\omega) = (\alpha, \beta)$ .

Surjectivity of $s$ : Let $\gamma \in \Omega^k(U \cap V)$ and $\{\rho_U, \rho_V\}$ be a partition of unity subordinate to $\{U, V\}$ . Set $\alpha = \rho_V \gamma$ (extended by zero outside $U \cap V$ , smooth on $U$ ) and $\beta = -\rho_U \gamma$ (smooth on $V$ ). Then $s(\alpha, \beta) = \rho_V \gamma + \rho_U \gamma = \gamma$ .

Step 2: Long exact sequence. The short exact sequence of cochain complexes induces a long exact sequence in cohomology by the snake lemma (zig-zag lemma). This gives the Mayer-Vietoris sequence.

Step 3: Connecting homomorphism. We describe $\delta$ explicitly. Given a closed $k$ -form $\gamma$ on $U \cap V$ , lift it: set $\alpha = \rho_V \gamma$ on $U$ and $\beta = -\rho_U \gamma$ on $V$ . Then $d\alpha$ on $U$ and $d\beta$ on $V$ agree on $U \cap V$ :

d\alpha|_{U \cap V} = d(\rho_V \gamma)|_{U \cap V} = d\rho_V \wedge \gamma + \rho_V \cdot 0 = d\rho_V \wedge \gamma,

d\beta|_{U \cap V} = -d(\rho_U \gamma)|_{U \cap V} = -d\rho_U \wedge \gamma = d\rho_V \wedge \gamma,

(since $d\rho_U + d\rho_V = 0$ ). So $d\alpha$ and $d\beta$ patch to a global closed $(k+1)$ -form $\Gamma$ on $M$ , and $\delta[\gamma] = [\Gamma]$ .

Step 4: Exactness verification. The exactness at each position follows from standard homological algebra (snake lemma). We verify:

At $H^k(U) \oplus H^k(V)$ : $s \circ r = 0$ (immediate), and $\ker s \subset \operatorname{im} r$ follows from patching.
At $H^k(U \cap V)$ : $\delta \circ s = 0$ because if $\gamma = \alpha|_{U \cap V} - \beta|_{U \cap V}$ with $d\alpha = 0, d\beta = 0$ , then the lift can be chosen with $d = 0$ .
At $H^{k+1}(M)$ : $r \circ \delta = 0$ because $\delta[\gamma]|_U = [d\alpha]$ which is exact. $\blacksquare$

■

Applications

RemarkComputational strategy

To compute $H^*(M)$ via Mayer-Vietoris: choose $U, V$ so that $H^*(U)$ , $H^*(V)$ , and $H^*(U \cap V)$ are known (or simpler). Then the long exact sequence, combined with knowledge of the maps, determines $H^*(M)$ . The key difficulty is usually computing the connecting homomorphism $\delta$ .

Example$H^*(S^1)$ via Mayer-Vietoris

Cover $S^1$ by two overlapping arcs $U, V \cong \mathbb{R}$ , with $U \cap V \cong \mathbb{R} \sqcup \mathbb{R}$ . The sequence gives $0 \to H^0(S^1) \to \mathbb{R}^2 \to \mathbb{R}^2 \xrightarrow{\delta} H^1(S^1) \to 0$ . The map $\mathbb{R}^2 \to \mathbb{R}^2$ sends $(a,b) \mapsto (a-b, a-b)$ , which has rank 1. So $H^0(S^1) \cong \mathbb{R}$ and $H^1(S^1) \cong \mathbb{R}$ .