De Rham Cohomology - Core Definitions

De Rham cohomology captures the global topological information of a manifold through the algebraic study of differential forms. It measures the failure of closed forms to be exact, providing computable topological invariants.

The De Rham Complex

Definition5.1De Rham complex

The de Rham complex of a smooth manifold $M$ is the cochain complex

0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \Omega^2(M) \xrightarrow{d} \cdots \xrightarrow{d} \Omega^n(M) \to 0,

where $\Omega^k(M)$ is the vector space of smooth $k$ -forms and $d: \Omega^k(M) \to \Omega^{k+1}(M)$ is the exterior derivative. The fundamental property $d^2 = 0$ makes this a cochain complex.

Definition5.2Closed and exact forms

A $k$ -form $\omega$ is closed if $d\omega = 0$ , and exact if $\omega = d\eta$ for some $(k-1)$ -form $\eta$ . Since $d^2 = 0$ , every exact form is closed. The converse generally fails, and the obstruction is measured by cohomology.

De Rham Cohomology Groups

Definition5.3De Rham cohomology

The $k$ -th de Rham cohomology group of $M$ is the quotient vector space

H^k_{\mathrm{dR}}(M) = \frac{\ker(d: \Omega^k \to \Omega^{k+1})}{\operatorname{im}(d: \Omega^{k-1} \to \Omega^k)} = \frac{Z^k(M)}{B^k(M)},

where $Z^k(M)$ denotes closed $k$ -forms and $B^k(M)$ denotes exact $k$ -forms. The equivalence class $[\omega] \in H^k_{\mathrm{dR}}(M)$ of a closed form $\omega$ is its cohomology class.

ExampleBasic computations

$H^0_{\mathrm{dR}}(M) \cong \mathbb{R}^c$ where $c$ is the number of connected components (closed 0-forms are locally constant functions).
$H^k_{\mathrm{dR}}(\mathbb{R}^n) = 0$ for $k \geq 1$ (the Poincare lemma: every closed form on $\mathbb{R}^n$ is exact).
$H^1_{\mathrm{dR}}(S^1) \cong \mathbb{R}$ , generated by $[d\theta]$ , reflecting the non-trivial loop.
$H^k_{\mathrm{dR}}(S^n) \cong \mathbb{R}$ for $k = 0, n$ and $0$ otherwise.

Functoriality

Definition5.4Pullback on cohomology

A smooth map $f: M \to N$ induces a pullback $f^*: H^k_{\mathrm{dR}}(N) \to H^k_{\mathrm{dR}}(M)$ defined by $f^*[\omega] = [f^*\omega]$ . This is well-defined because $f^*$ commutes with $d$ : if $d\omega = 0$ then $d(f^*\omega) = f^*(d\omega) = 0$ , and if $\omega = d\eta$ then $f^*\omega = d(f^*\eta)$ .

RemarkHomotopy invariance

If $f, g: M \to N$ are smoothly homotopic, then $f^* = g^*: H^k_{\mathrm{dR}}(N) \to H^k_{\mathrm{dR}}(M)$ . This is proved using the chain homotopy operator and implies that de Rham cohomology is a homotopy invariant: homotopy equivalent manifolds have isomorphic cohomology.

ExampleContractible manifolds

Since any contractible manifold is homotopy equivalent to a point, its de Rham cohomology is $H^0 \cong \mathbb{R}$ and $H^k = 0$ for $k \geq 1$ . This gives a conceptual proof of the Poincare lemma for star-shaped domains.