Mayer-Vietoris and Computations

The Mayer-Vietoris sequence is the primary computational tool for de Rham cohomology, allowing one to compute the cohomology of complicated spaces by decomposing them into simpler pieces.

The Mayer-Vietoris Sequence

Theorem5.2Mayer-Vietoris sequence

Let $M = U \cup V$ where $U, V$ are open subsets. There is a long exact sequence in de Rham cohomology:

\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

where $r[\omega] = ([\omega|_U], [\omega|_V])$ is the restriction map, $s([\alpha], [\beta]) = [\alpha|_{U \cap V} - \beta|_{U \cap V}]$ , and $\delta$ is the connecting homomorphism.

DefinitionConnecting homomorphism

The connecting homomorphism $\delta: H^k(U \cap V) \to H^{k+1}(M)$ is constructed as follows. Given a closed $k$ -form $\omega$ on $U \cap V$ , choose a partition of unity $\{\rho_U, \rho_V\}$ subordinate to $\{U, V\}$ . Define $\eta_V = d(\rho_U \omega)$ on $V$ (extended by zero) and $\eta_U = -d(\rho_V \omega)$ on $U$ . These agree on $U \cap V$ and patch to a closed $(k+1)$ -form on $M$ , giving $\delta[\omega]$ .

Computations

ExampleCohomology of the torus $T^2$

Write $T^2 = U \cup V$ where $U$ and $V$ are open cylinders overlapping in two disjoint annuli, so $U \cap V \simeq S^1 \sqcup S^1$ . The Mayer-Vietoris sequence gives:

H^0(T^2) \cong \mathbb{R}, \quad H^1(T^2) \cong \mathbb{R}^2, \quad H^2(T^2) \cong \mathbb{R}.

The generators of $H^1$ correspond to the two independent loops (meridian and longitude), and the generator of $H^2$ is the area form.

ExampleCohomology of $S^n$ by induction

Cover $S^n$ by $U = S^n \setminus \{N\}$ and $V = S^n \setminus \{S\}$ (removing north and south poles). Both are contractible ( $\cong \mathbb{R}^n$ ) and $U \cap V \simeq S^{n-1}$ . The Mayer-Vietoris sequence gives:

For $k \geq 2$ : $H^k(S^n) \cong H^{k-1}(S^{n-1})$ . By induction: $H^k(S^n) = \mathbb{R}$ if $k = 0$ or $n$ , and $0$ otherwise.

The Kunneth Formula

Theorem5.3Kunneth formula

For smooth manifolds $M$ and $N$ with $H^*(M)$ finite-dimensional,

H^k_{\mathrm{dR}}(M \times N) \cong \bigoplus_{p+q=k} H^p_{\mathrm{dR}}(M) \otimes H^q_{\mathrm{dR}}(N).

ExampleKunneth for the torus

$T^n = (S^1)^n$ . By Kunneth, $H^k(T^n) \cong \binom{n}{k} \mathbb{R}$ , so the Betti numbers are binomial coefficients: $b_k = \binom{n}{k}$ . The total dimension is $\sum b_k = 2^n$ .

RemarkRing structure

The wedge product $\wedge: \Omega^p(M) \times \Omega^q(M) \to \Omega^{p+q}(M)$ descends to cohomology, giving $H^*_{\mathrm{dR}}(M)$ the structure of a graded-commutative algebra. This cup product structure carries more information than the individual cohomology groups alone.